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Metamath Proof Explorer

Theorem icccncfext

Description: A continuous function on a closed interval can be extended to a continuous function on the whole real line. (Contributed by Glauco Siliprandi, 11-Dec-2019)

Ref Expression
Hypotheses icccncfext.1 ⊢ Ⅎ 𝑥 𝐹
icccncfext.2 ⊢ 𝐽 = ( topGen ‘ ran (,) )
icccncfext.3 ⊢ 𝑌 = ∪ 𝐾
icccncfext.4 ⊢ 𝐺 = ( 𝑥 ∈ ℝ ↦ if ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑥 ) , if ( 𝑥 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) )
icccncfext.5 ⊢ ( 𝜑 → 𝐴 ∈ ℝ )
icccncfext.6 ⊢ ( 𝜑 → 𝐵 ∈ ℝ )
icccncfext.7 ⊢ ( 𝜑 → 𝐴 ≤ 𝐵 )
icccncfext.8 ⊢ ( 𝜑 → 𝐾 ∈ Top )
icccncfext.9 ⊢ ( 𝜑 → 𝐹 ∈ ( ( 𝐽 ↾t ( 𝐴 [,] 𝐵 ) ) Cn 𝐾 ) )
Assertion icccncfext ( 𝜑 → ( 𝐺 ∈ ( 𝐽 Cn ( 𝐾 ↾t ran 𝐹 ) ) ∧ ( 𝐺 ↾ ( 𝐴 [,] 𝐵 ) ) = 𝐹 ) )

Proof

Step Hyp Ref Expression
1 icccncfext.1 ⊢ Ⅎ 𝑥 𝐹
2 icccncfext.2 ⊢ 𝐽 = ( topGen ‘ ran (,) )
3 icccncfext.3 ⊢ 𝑌 = ∪ 𝐾
4 icccncfext.4 ⊢ 𝐺 = ( 𝑥 ∈ ℝ ↦ if ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑥 ) , if ( 𝑥 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) )
5 icccncfext.5 ⊢ ( 𝜑 → 𝐴 ∈ ℝ )
6 icccncfext.6 ⊢ ( 𝜑 → 𝐵 ∈ ℝ )
7 icccncfext.7 ⊢ ( 𝜑 → 𝐴 ≤ 𝐵 )
8 icccncfext.8 ⊢ ( 𝜑 → 𝐾 ∈ Top )
9 icccncfext.9 ⊢ ( 𝜑 → 𝐹 ∈ ( ( 𝐽 ↾t ( 𝐴 [,] 𝐵 ) ) Cn 𝐾 ) )
10 retopon ⊢ ( topGen ‘ ran (,) ) ∈ ( TopOn ‘ ℝ )
11 2 10 eqeltri ⊢ 𝐽 ∈ ( TopOn ‘ ℝ )
12 5 6 iccssred ⊢ ( 𝜑 → ( 𝐴 [,] 𝐵 ) ⊆ ℝ )
13 resttopon ⊢ ( ( 𝐽 ∈ ( TopOn ‘ ℝ ) ∧ ( 𝐴 [,] 𝐵 ) ⊆ ℝ ) → ( 𝐽 ↾t ( 𝐴 [,] 𝐵 ) ) ∈ ( TopOn ‘ ( 𝐴 [,] 𝐵 ) ) )
14 11 12 13 sylancr ⊢ ( 𝜑 → ( 𝐽 ↾t ( 𝐴 [,] 𝐵 ) ) ∈ ( TopOn ‘ ( 𝐴 [,] 𝐵 ) ) )
15 8 3 jctir ⊢ ( 𝜑 → ( 𝐾 ∈ Top ∧ 𝑌 = ∪ 𝐾 ) )
16 istopon ⊢ ( 𝐾 ∈ ( TopOn ‘ 𝑌 ) ↔ ( 𝐾 ∈ Top ∧ 𝑌 = ∪ 𝐾 ) )
17 15 16 sylibr ⊢ ( 𝜑 → 𝐾 ∈ ( TopOn ‘ 𝑌 ) )
18 cnf2 ⊢ ( ( ( 𝐽 ↾t ( 𝐴 [,] 𝐵 ) ) ∈ ( TopOn ‘ ( 𝐴 [,] 𝐵 ) ) ∧ 𝐾 ∈ ( TopOn ‘ 𝑌 ) ∧ 𝐹 ∈ ( ( 𝐽 ↾t ( 𝐴 [,] 𝐵 ) ) Cn 𝐾 ) ) → 𝐹 : ( 𝐴 [,] 𝐵 ) ⟶ 𝑌 )
19 14 17 9 18 syl3anc ⊢ ( 𝜑 → 𝐹 : ( 𝐴 [,] 𝐵 ) ⟶ 𝑌 )
20 19 ffnd ⊢ ( 𝜑 → 𝐹 Fn ( 𝐴 [,] 𝐵 ) )
21 dffn3 ⊢ ( 𝐹 Fn ( 𝐴 [,] 𝐵 ) ↔ 𝐹 : ( 𝐴 [,] 𝐵 ) ⟶ ran 𝐹 )
22 20 21 sylib ⊢ ( 𝜑 → 𝐹 : ( 𝐴 [,] 𝐵 ) ⟶ ran 𝐹 )
23 22 ffvelcdmda ⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝐹 ‘ 𝑦 ) ∈ ran 𝐹 )
24 fnfun ⊢ ( 𝐹 Fn ( 𝐴 [,] 𝐵 ) → Fun 𝐹 )
25 20 24 syl ⊢ ( 𝜑 → Fun 𝐹 )
26 5 rexrd ⊢ ( 𝜑 → 𝐴 ∈ ℝ* )
27 6 rexrd ⊢ ( 𝜑 → 𝐵 ∈ ℝ* )
28 lbicc2 ⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵 ) → 𝐴 ∈ ( 𝐴 [,] 𝐵 ) )
29 26 27 7 28 syl3anc ⊢ ( 𝜑 → 𝐴 ∈ ( 𝐴 [,] 𝐵 ) )
30 20 fndmd ⊢ ( 𝜑 → dom 𝐹 = ( 𝐴 [,] 𝐵 ) )
31 30 eqcomd ⊢ ( 𝜑 → ( 𝐴 [,] 𝐵 ) = dom 𝐹 )
32 29 31 eleqtrd ⊢ ( 𝜑 → 𝐴 ∈ dom 𝐹 )
33 fvelrn ⊢ ( ( Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹 ) → ( 𝐹 ‘ 𝐴 ) ∈ ran 𝐹 )
34 25 32 33 syl2anc ⊢ ( 𝜑 → ( 𝐹 ‘ 𝐴 ) ∈ ran 𝐹 )
35 ubicc2 ⊢ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝐴 ≤ 𝐵 ) → 𝐵 ∈ ( 𝐴 [,] 𝐵 ) )
36 26 27 7 35 syl3anc ⊢ ( 𝜑 → 𝐵 ∈ ( 𝐴 [,] 𝐵 ) )
37 36 31 eleqtrd ⊢ ( 𝜑 → 𝐵 ∈ dom 𝐹 )
38 fvelrn ⊢ ( ( Fun 𝐹 ∧ 𝐵 ∈ dom 𝐹 ) → ( 𝐹 ‘ 𝐵 ) ∈ ran 𝐹 )
39 25 37 38 syl2anc ⊢ ( 𝜑 → ( 𝐹 ‘ 𝐵 ) ∈ ran 𝐹 )
40 34 39 ifcld ⊢ ( 𝜑 → if ( 𝑦 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ∈ ran 𝐹 )
41 40 adantr ⊢ ( ( 𝜑 ∧ ¬ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → if ( 𝑦 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ∈ ran 𝐹 )
42 23 41 ifclda ⊢ ( 𝜑 → if ( 𝑦 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑦 ) , if ( 𝑦 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) ∈ ran 𝐹 )
43 42 adantr ⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → if ( 𝑦 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑦 ) , if ( 𝑦 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) ∈ ran 𝐹 )
44 nfv ⊢ Ⅎ 𝑦 𝑥 ∈ ( 𝐴 [,] 𝐵 )
45 nfcv ⊢ Ⅎ 𝑦 ( 𝐹 ‘ 𝑥 )
46 nfcv ⊢ Ⅎ 𝑦 if ( 𝑥 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) )
47 44 45 46 nfif ⊢ Ⅎ 𝑦 if ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑥 ) , if ( 𝑥 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) )
48 nfv ⊢ Ⅎ 𝑥 𝑦 ∈ ( 𝐴 [,] 𝐵 )
49 nfcv ⊢ Ⅎ 𝑥 𝑦
50 1 49 nffv ⊢ Ⅎ 𝑥 ( 𝐹 ‘ 𝑦 )
51 nfv ⊢ Ⅎ 𝑥 𝑦 < 𝐴
52 nfcv ⊢ Ⅎ 𝑥 𝐴
53 1 52 nffv ⊢ Ⅎ 𝑥 ( 𝐹 ‘ 𝐴 )
54 nfcv ⊢ Ⅎ 𝑥 𝐵
55 1 54 nffv ⊢ Ⅎ 𝑥 ( 𝐹 ‘ 𝐵 )
56 51 53 55 nfif ⊢ Ⅎ 𝑥 if ( 𝑦 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) )
57 48 50 56 nfif ⊢ Ⅎ 𝑥 if ( 𝑦 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑦 ) , if ( 𝑦 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) )
58 eleq1 ⊢ ( 𝑥 = 𝑦 → ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) ↔ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) )
59 fveq2 ⊢ ( 𝑥 = 𝑦 → ( 𝐹 ‘ 𝑥 ) = ( 𝐹 ‘ 𝑦 ) )
60 breq1 ⊢ ( 𝑥 = 𝑦 → ( 𝑥 < 𝐴 ↔ 𝑦 < 𝐴 ) )
61 60 ifbid ⊢ ( 𝑥 = 𝑦 → if ( 𝑥 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) = if ( 𝑦 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) )
62 58 59 61 ifbieq12d ⊢ ( 𝑥 = 𝑦 → if ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑥 ) , if ( 𝑥 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) = if ( 𝑦 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑦 ) , if ( 𝑦 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) )
63 47 57 62 cbvmpt ⊢ ( 𝑥 ∈ ℝ ↦ if ( 𝑥 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑥 ) , if ( 𝑥 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) ) = ( 𝑦 ∈ ℝ ↦ if ( 𝑦 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑦 ) , if ( 𝑦 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) )
64 4 63 eqtri ⊢ 𝐺 = ( 𝑦 ∈ ℝ ↦ if ( 𝑦 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑦 ) , if ( 𝑦 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) )
65 43 64 fmptd ⊢ ( 𝜑 → 𝐺 : ℝ ⟶ ran 𝐹 )
66 65 adantr ⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → 𝐺 : ℝ ⟶ ran 𝐹 )
67 simplll ⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑢 ∈ ( 𝐾 ↾t ran 𝐹 ) ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) → 𝜑 )
68 simplr ⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑢 ∈ ( 𝐾 ↾t ran 𝐹 ) ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) → 𝑢 ∈ ( 𝐾 ↾t ran 𝐹 ) )
69 67 68 jca ⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑢 ∈ ( 𝐾 ↾t ran 𝐹 ) ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) → ( 𝜑 ∧ 𝑢 ∈ ( 𝐾 ↾t ran 𝐹 ) ) )
70 ssidd ⊢ ( 𝜑 → ran 𝐹 ⊆ ran 𝐹 )
71 19 frnd ⊢ ( 𝜑 → ran 𝐹 ⊆ 𝑌 )
72 cnrest2 ⊢ ( ( 𝐾 ∈ ( TopOn ‘ 𝑌 ) ∧ ran 𝐹 ⊆ ran 𝐹 ∧ ran 𝐹 ⊆ 𝑌 ) → ( 𝐹 ∈ ( ( 𝐽 ↾t ( 𝐴 [,] 𝐵 ) ) Cn 𝐾 ) ↔ 𝐹 ∈ ( ( 𝐽 ↾t ( 𝐴 [,] 𝐵 ) ) Cn ( 𝐾 ↾t ran 𝐹 ) ) ) )
73 17 70 71 72 syl3anc ⊢ ( 𝜑 → ( 𝐹 ∈ ( ( 𝐽 ↾t ( 𝐴 [,] 𝐵 ) ) Cn 𝐾 ) ↔ 𝐹 ∈ ( ( 𝐽 ↾t ( 𝐴 [,] 𝐵 ) ) Cn ( 𝐾 ↾t ran 𝐹 ) ) ) )
74 9 73 mpbid ⊢ ( 𝜑 → 𝐹 ∈ ( ( 𝐽 ↾t ( 𝐴 [,] 𝐵 ) ) Cn ( 𝐾 ↾t ran 𝐹 ) ) )
75 74 anim1i ⊢ ( ( 𝜑 ∧ 𝑢 ∈ ( 𝐾 ↾t ran 𝐹 ) ) → ( 𝐹 ∈ ( ( 𝐽 ↾t ( 𝐴 [,] 𝐵 ) ) Cn ( 𝐾 ↾t ran 𝐹 ) ) ∧ 𝑢 ∈ ( 𝐾 ↾t ran 𝐹 ) ) )
76 cnima ⊢ ( ( 𝐹 ∈ ( ( 𝐽 ↾t ( 𝐴 [,] 𝐵 ) ) Cn ( 𝐾 ↾t ran 𝐹 ) ) ∧ 𝑢 ∈ ( 𝐾 ↾t ran 𝐹 ) ) → ( ◡ 𝐹 “ 𝑢 ) ∈ ( 𝐽 ↾t ( 𝐴 [,] 𝐵 ) ) )
77 69 75 76 3syl ⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑢 ∈ ( 𝐾 ↾t ran 𝐹 ) ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) → ( ◡ 𝐹 “ 𝑢 ) ∈ ( 𝐽 ↾t ( 𝐴 [,] 𝐵 ) ) )
78 retop ⊢ ( topGen ‘ ran (,) ) ∈ Top
79 2 78 eqeltri ⊢ 𝐽 ∈ Top
80 79 a1i ⊢ ( 𝜑 → 𝐽 ∈ Top )
81 reex ⊢ ℝ ∈ V
82 81 a1i ⊢ ( 𝜑 → ℝ ∈ V )
83 82 12 ssexd ⊢ ( 𝜑 → ( 𝐴 [,] 𝐵 ) ∈ V )
84 80 83 jca ⊢ ( 𝜑 → ( 𝐽 ∈ Top ∧ ( 𝐴 [,] 𝐵 ) ∈ V ) )
85 67 84 syl ⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑢 ∈ ( 𝐾 ↾t ran 𝐹 ) ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) → ( 𝐽 ∈ Top ∧ ( 𝐴 [,] 𝐵 ) ∈ V ) )
86 elrest ⊢ ( ( 𝐽 ∈ Top ∧ ( 𝐴 [,] 𝐵 ) ∈ V ) → ( ( ◡ 𝐹 “ 𝑢 ) ∈ ( 𝐽 ↾t ( 𝐴 [,] 𝐵 ) ) ↔ ∃ 𝑤 ∈ 𝐽 ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) )
87 85 86 syl ⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑢 ∈ ( 𝐾 ↾t ran 𝐹 ) ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) → ( ( ◡ 𝐹 “ 𝑢 ) ∈ ( 𝐽 ↾t ( 𝐴 [,] 𝐵 ) ) ↔ ∃ 𝑤 ∈ 𝐽 ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) )
88 77 87 mpbid ⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑢 ∈ ( 𝐾 ↾t ran 𝐹 ) ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) → ∃ 𝑤 ∈ 𝐽 ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) )
89 67 3ad2ant1 ⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑢 ∈ ( 𝐾 ↾t ran 𝐹 ) ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) → 𝜑 )
90 simpllr ⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑢 ∈ ( 𝐾 ↾t ran 𝐹 ) ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) → 𝑦 ∈ ℝ )
91 90 3ad2ant1 ⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑢 ∈ ( 𝐾 ↾t ran 𝐹 ) ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) → 𝑦 ∈ ℝ )
92 simp1r ⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑢 ∈ ( 𝐾 ↾t ran 𝐹 ) ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) → ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 )
93 89 91 92 jca31 ⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑢 ∈ ( 𝐾 ↾t ran 𝐹 ) ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) → ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) )
94 simpll2 ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → 𝑤 ∈ 𝐽 )
95 iooretop ⊢ ( -∞ (,) 𝐴 ) ∈ ( topGen ‘ ran (,) )
96 95 2 eleqtrri ⊢ ( -∞ (,) 𝐴 ) ∈ 𝐽
97 iooretop ⊢ ( 𝐵 (,) +∞ ) ∈ ( topGen ‘ ran (,) )
98 97 2 eleqtrri ⊢ ( 𝐵 (,) +∞ ) ∈ 𝐽
99 unopn ⊢ ( ( 𝐽 ∈ Top ∧ ( -∞ (,) 𝐴 ) ∈ 𝐽 ∧ ( 𝐵 (,) +∞ ) ∈ 𝐽 ) → ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ∈ 𝐽 )
100 79 96 98 99 mp3an ⊢ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ∈ 𝐽
101 unopn ⊢ ( ( 𝐽 ∈ Top ∧ 𝑤 ∈ 𝐽 ∧ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ∈ 𝐽 ) → ( 𝑤 ∪ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) ∈ 𝐽 )
102 79 100 101 mp3an13 ⊢ ( 𝑤 ∈ 𝐽 → ( 𝑤 ∪ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) ∈ 𝐽 )
103 94 102 syl ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( 𝑤 ∪ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) ∈ 𝐽 )
104 simpl1l ⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) → ( 𝜑 ∧ 𝑦 ∈ ℝ ) )
105 104 adantr ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( 𝜑 ∧ 𝑦 ∈ ℝ ) )
106 simpl1r ⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) → ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 )
107 106 adantr ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 )
108 simpll3 ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) )
109 difreicc ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( ℝ ∖ ( 𝐴 [,] 𝐵 ) ) = ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) )
110 5 6 109 syl2anc ⊢ ( 𝜑 → ( ℝ ∖ ( 𝐴 [,] 𝐵 ) ) = ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) )
111 110 eqcomd ⊢ ( 𝜑 → ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) = ( ℝ ∖ ( 𝐴 [,] 𝐵 ) ) )
112 111 eleq2d ⊢ ( 𝜑 → ( 𝑦 ∈ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ↔ 𝑦 ∈ ( ℝ ∖ ( 𝐴 [,] 𝐵 ) ) ) )
113 112 notbid ⊢ ( 𝜑 → ( ¬ 𝑦 ∈ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ↔ ¬ 𝑦 ∈ ( ℝ ∖ ( 𝐴 [,] 𝐵 ) ) ) )
114 113 biimpa ⊢ ( ( 𝜑 ∧ ¬ 𝑦 ∈ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) → ¬ 𝑦 ∈ ( ℝ ∖ ( 𝐴 [,] 𝐵 ) ) )
115 eldif ⊢ ( 𝑦 ∈ ( ℝ ∖ ( 𝐴 [,] 𝐵 ) ) ↔ ( 𝑦 ∈ ℝ ∧ ¬ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) )
116 114 115 sylnib ⊢ ( ( 𝜑 ∧ ¬ 𝑦 ∈ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) → ¬ ( 𝑦 ∈ ℝ ∧ ¬ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) )
117 imnan ⊢ ( ( 𝑦 ∈ ℝ → ¬ ¬ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) ↔ ¬ ( 𝑦 ∈ ℝ ∧ ¬ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) )
118 116 117 sylibr ⊢ ( ( 𝜑 ∧ ¬ 𝑦 ∈ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) → ( 𝑦 ∈ ℝ → ¬ ¬ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) )
119 118 imp ⊢ ( ( ( 𝜑 ∧ ¬ 𝑦 ∈ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) ∧ 𝑦 ∈ ℝ ) → ¬ ¬ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) )
120 119 notnotrd ⊢ ( ( ( 𝜑 ∧ ¬ 𝑦 ∈ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) ∧ 𝑦 ∈ ℝ ) → 𝑦 ∈ ( 𝐴 [,] 𝐵 ) )
121 120 an32s ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) → 𝑦 ∈ ( 𝐴 [,] 𝐵 ) )
122 121 adantlr ⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) → 𝑦 ∈ ( 𝐴 [,] 𝐵 ) )
123 simplll ⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) → 𝜑 )
124 12 sselda ⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝑦 ∈ ℝ )
125 19 adantr ⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝐹 : ( 𝐴 [,] 𝐵 ) ⟶ 𝑌 )
126 125 ffvelcdmda ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝐹 ‘ 𝑦 ) ∈ 𝑌 )
127 19 29 ffvelcdmd ⊢ ( 𝜑 → ( 𝐹 ‘ 𝐴 ) ∈ 𝑌 )
128 127 ad3antrrr ⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) ∧ ¬ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) ∧ 𝑦 < 𝐴 ) → ( 𝐹 ‘ 𝐴 ) ∈ 𝑌 )
129 19 36 ffvelcdmd ⊢ ( 𝜑 → ( 𝐹 ‘ 𝐵 ) ∈ 𝑌 )
130 129 ad3antrrr ⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) ∧ ¬ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) ∧ ¬ 𝑦 < 𝐴 ) → ( 𝐹 ‘ 𝐵 ) ∈ 𝑌 )
131 128 130 ifclda ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) ∧ ¬ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → if ( 𝑦 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ∈ 𝑌 )
132 126 131 ifclda ⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → if ( 𝑦 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑦 ) , if ( 𝑦 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) ∈ 𝑌 )
133 64 fvmpt2 ⊢ ( ( 𝑦 ∈ ℝ ∧ if ( 𝑦 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑦 ) , if ( 𝑦 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) ∈ 𝑌 ) → ( 𝐺 ‘ 𝑦 ) = if ( 𝑦 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑦 ) , if ( 𝑦 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) )
134 124 132 133 syl2anc ⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝐺 ‘ 𝑦 ) = if ( 𝑦 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑦 ) , if ( 𝑦 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) )
135 simpr ⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝑦 ∈ ( 𝐴 [,] 𝐵 ) )
136 135 iftrued ⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → if ( 𝑦 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑦 ) , if ( 𝑦 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) = ( 𝐹 ‘ 𝑦 ) )
137 134 136 eqtrd ⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝐺 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑦 ) )
138 137 eqcomd ⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝐹 ‘ 𝑦 ) = ( 𝐺 ‘ 𝑦 ) )
139 123 122 138 syl2anc ⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) → ( 𝐹 ‘ 𝑦 ) = ( 𝐺 ‘ 𝑦 ) )
140 simplr ⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) → ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 )
141 139 140 eqeltrd ⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) → ( 𝐹 ‘ 𝑦 ) ∈ 𝑢 )
142 123 20 syl ⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) → 𝐹 Fn ( 𝐴 [,] 𝐵 ) )
143 elpreima ⊢ ( 𝐹 Fn ( 𝐴 [,] 𝐵 ) → ( 𝑦 ∈ ( ◡ 𝐹 “ 𝑢 ) ↔ ( 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ∧ ( 𝐹 ‘ 𝑦 ) ∈ 𝑢 ) ) )
144 142 143 syl ⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) → ( 𝑦 ∈ ( ◡ 𝐹 “ 𝑢 ) ↔ ( 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ∧ ( 𝐹 ‘ 𝑦 ) ∈ 𝑢 ) ) )
145 122 141 144 mpbir2and ⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) → 𝑦 ∈ ( ◡ 𝐹 “ 𝑢 ) )
146 145 adantlr ⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ 𝑦 ∈ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) → 𝑦 ∈ ( ◡ 𝐹 “ 𝑢 ) )
147 simplr ⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ 𝑦 ∈ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) → ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) )
148 146 147 eleqtrd ⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ 𝑦 ∈ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) → 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) )
149 elin ⊢ ( 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ↔ ( 𝑦 ∈ 𝑤 ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) )
150 148 149 sylib ⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ 𝑦 ∈ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) → ( 𝑦 ∈ 𝑤 ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) )
151 150 simpld ⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ 𝑦 ∈ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) → 𝑦 ∈ 𝑤 )
152 151 ex ⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) → ( ¬ 𝑦 ∈ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) → 𝑦 ∈ 𝑤 ) )
153 152 orrd ⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) → ( 𝑦 ∈ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ∨ 𝑦 ∈ 𝑤 ) )
154 153 orcomd ⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) → ( 𝑦 ∈ 𝑤 ∨ 𝑦 ∈ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) )
155 elun ⊢ ( 𝑦 ∈ ( 𝑤 ∪ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) ↔ ( 𝑦 ∈ 𝑤 ∨ 𝑦 ∈ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) )
156 154 155 sylibr ⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) → 𝑦 ∈ ( 𝑤 ∪ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) )
157 105 107 108 156 syl21anc ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → 𝑦 ∈ ( 𝑤 ∪ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) )
158 imaundi ⊢ ( 𝐺 “ ( 𝑤 ∪ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) ) = ( ( 𝐺 “ 𝑤 ) ∪ ( 𝐺 “ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) )
159 105 simpld ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → 𝜑 )
160 toponss ⊢ ( ( 𝐽 ∈ ( TopOn ‘ ℝ ) ∧ 𝑤 ∈ 𝐽 ) → 𝑤 ⊆ ℝ )
161 11 94 160 sylancr ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → 𝑤 ⊆ ℝ )
162 159 161 108 jca31 ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) )
163 simplr ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 )
164 simpr ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 )
165 4 funmpt2 ⊢ Fun 𝐺
166 165 a1i ⊢ ( 𝜑 → Fun 𝐺 )
167 166 ad5antr ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑦 ∈ ( 𝐺 “ 𝑤 ) ) → Fun 𝐺 )
168 fvelima ⊢ ( ( Fun 𝐺 ∧ 𝑦 ∈ ( 𝐺 “ 𝑤 ) ) → ∃ 𝑧 ∈ 𝑤 ( 𝐺 ‘ 𝑧 ) = 𝑦 )
169 167 168 sylancom ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑦 ∈ ( 𝐺 “ 𝑤 ) ) → ∃ 𝑧 ∈ 𝑤 ( 𝐺 ‘ 𝑧 ) = 𝑦 )
170 eqcom ⊢ ( ( 𝐺 ‘ 𝑧 ) = 𝑦 ↔ 𝑦 = ( 𝐺 ‘ 𝑧 ) )
171 170 biimpi ⊢ ( ( 𝐺 ‘ 𝑧 ) = 𝑦 → 𝑦 = ( 𝐺 ‘ 𝑧 ) )
172 171 3ad2ant3 ⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑦 ∈ ( 𝐺 “ 𝑤 ) ) ∧ 𝑧 ∈ 𝑤 ∧ ( 𝐺 ‘ 𝑧 ) = 𝑦 ) → 𝑦 = ( 𝐺 ‘ 𝑧 ) )
173 simp1ll ⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑦 ∈ ( 𝐺 “ 𝑤 ) ) ∧ 𝑧 ∈ 𝑤 ∧ ( 𝐺 ‘ 𝑧 ) = 𝑦 ) → ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) )
174 simp1lr ⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑦 ∈ ( 𝐺 “ 𝑤 ) ) ∧ 𝑧 ∈ 𝑤 ∧ ( 𝐺 ‘ 𝑧 ) = 𝑦 ) → ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 )
175 simp2 ⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑦 ∈ ( 𝐺 “ 𝑤 ) ) ∧ 𝑧 ∈ 𝑤 ∧ ( 𝐺 ‘ 𝑧 ) = 𝑦 ) → 𝑧 ∈ 𝑤 )
176 simp-5l ⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑧 ∈ 𝑤 ) ∧ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝜑 ∧ 𝑤 ⊆ ℝ ) )
177 simp-5r ⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑧 ∈ 𝑤 ) ∧ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) )
178 simplr ⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑧 ∈ 𝑤 ) ∧ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝑧 ∈ 𝑤 )
179 176 177 178 jca31 ⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑧 ∈ 𝑤 ) ∧ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑧 ∈ 𝑤 ) )
180 eleq1 ⊢ ( 𝑦 = 𝑧 → ( 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ↔ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) )
181 180 anbi2d ⊢ ( 𝑦 = 𝑧 → ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) ↔ ( 𝜑 ∧ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) ) )
182 fveq2 ⊢ ( 𝑦 = 𝑧 → ( 𝐺 ‘ 𝑦 ) = ( 𝐺 ‘ 𝑧 ) )
183 fveq2 ⊢ ( 𝑦 = 𝑧 → ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑧 ) )
184 182 183 eqeq12d ⊢ ( 𝑦 = 𝑧 → ( ( 𝐺 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑦 ) ↔ ( 𝐺 ‘ 𝑧 ) = ( 𝐹 ‘ 𝑧 ) ) )
185 181 184 imbi12d ⊢ ( 𝑦 = 𝑧 → ( ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝐺 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑦 ) ) ↔ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝐺 ‘ 𝑧 ) = ( 𝐹 ‘ 𝑧 ) ) ) )
186 185 137 chvarvv ⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝐺 ‘ 𝑧 ) = ( 𝐹 ‘ 𝑧 ) )
187 186 ad4ant14 ⊢ ( ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑧 ∈ 𝑤 ) ∧ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝐺 ‘ 𝑧 ) = ( 𝐹 ‘ 𝑧 ) )
188 187 adantl3r ⊢ ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑧 ∈ 𝑤 ) ∧ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝐺 ‘ 𝑧 ) = ( 𝐹 ‘ 𝑧 ) )
189 simp-4l ⊢ ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑧 ∈ 𝑤 ) ∧ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝜑 )
190 simp-4r ⊢ ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑧 ∈ 𝑤 ) ∧ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝑤 ⊆ ℝ )
191 simplr ⊢ ( ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑧 ∈ 𝑤 ) ∧ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝑧 ∈ 𝑤 )
192 simpr ⊢ ( ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑧 ∈ 𝑤 ) ∧ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝑧 ∈ ( 𝐴 [,] 𝐵 ) )
193 191 192 elind ⊢ ( ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑧 ∈ 𝑤 ) ∧ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝑧 ∈ ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) )
194 eqcom ⊢ ( ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ↔ ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) = ( ◡ 𝐹 “ 𝑢 ) )
195 194 biimpi ⊢ ( ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) → ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) = ( ◡ 𝐹 “ 𝑢 ) )
196 195 ad3antlr ⊢ ( ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑧 ∈ 𝑤 ) ∧ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) = ( ◡ 𝐹 “ 𝑢 ) )
197 193 196 eleqtrd ⊢ ( ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑧 ∈ 𝑤 ) ∧ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝑧 ∈ ( ◡ 𝐹 “ 𝑢 ) )
198 197 adantl3r ⊢ ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑧 ∈ 𝑤 ) ∧ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝑧 ∈ ( ◡ 𝐹 “ 𝑢 ) )
199 simpr ⊢ ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ 𝑧 ∈ ( ◡ 𝐹 “ 𝑢 ) ) → 𝑧 ∈ ( ◡ 𝐹 “ 𝑢 ) )
200 20 ad2antrr ⊢ ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ 𝑧 ∈ ( ◡ 𝐹 “ 𝑢 ) ) → 𝐹 Fn ( 𝐴 [,] 𝐵 ) )
201 elpreima ⊢ ( 𝐹 Fn ( 𝐴 [,] 𝐵 ) → ( 𝑧 ∈ ( ◡ 𝐹 “ 𝑢 ) ↔ ( 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ∧ ( 𝐹 ‘ 𝑧 ) ∈ 𝑢 ) ) )
202 200 201 syl ⊢ ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ 𝑧 ∈ ( ◡ 𝐹 “ 𝑢 ) ) → ( 𝑧 ∈ ( ◡ 𝐹 “ 𝑢 ) ↔ ( 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ∧ ( 𝐹 ‘ 𝑧 ) ∈ 𝑢 ) ) )
203 199 202 mpbid ⊢ ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ 𝑧 ∈ ( ◡ 𝐹 “ 𝑢 ) ) → ( 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ∧ ( 𝐹 ‘ 𝑧 ) ∈ 𝑢 ) )
204 203 simprd ⊢ ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ 𝑧 ∈ ( ◡ 𝐹 “ 𝑢 ) ) → ( 𝐹 ‘ 𝑧 ) ∈ 𝑢 )
205 189 190 198 204 syl21anc ⊢ ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑧 ∈ 𝑤 ) ∧ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝐹 ‘ 𝑧 ) ∈ 𝑢 )
206 188 205 eqeltrd ⊢ ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑧 ∈ 𝑤 ) ∧ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝐺 ‘ 𝑧 ) ∈ 𝑢 )
207 179 206 sylancom ⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑧 ∈ 𝑤 ) ∧ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝐺 ‘ 𝑧 ) ∈ 𝑢 )
208 simp-5l ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑧 ∈ 𝑤 ) ∧ ¬ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝜑 )
209 simp-4r ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑧 ∈ 𝑤 ) ∧ ¬ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 )
210 208 209 jca ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑧 ∈ 𝑤 ) ∧ ¬ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝜑 ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) )
211 simpllr ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑧 ∈ 𝑤 ) ∧ ¬ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 )
212 simp-5r ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑧 ∈ 𝑤 ) ∧ ¬ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝑤 ⊆ ℝ )
213 simplr ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑧 ∈ 𝑤 ) ∧ ¬ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝑧 ∈ 𝑤 )
214 212 213 sseldd ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑧 ∈ 𝑤 ) ∧ ¬ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝑧 ∈ ℝ )
215 210 211 214 jca31 ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑧 ∈ 𝑤 ) ∧ ¬ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → ( ( ( 𝜑 ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑧 ∈ ℝ ) )
216 64 a1i ⊢ ( ( ( ( ( 𝜑 ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑧 ∈ ℝ ) ∧ ¬ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝐺 = ( 𝑦 ∈ ℝ ↦ if ( 𝑦 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑦 ) , if ( 𝑦 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) ) )
217 breq1 ⊢ ( 𝑦 = 𝑧 → ( 𝑦 < 𝐴 ↔ 𝑧 < 𝐴 ) )
218 217 ifbid ⊢ ( 𝑦 = 𝑧 → if ( 𝑦 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) = if ( 𝑧 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) )
219 180 183 218 ifbieq12d ⊢ ( 𝑦 = 𝑧 → if ( 𝑦 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑦 ) , if ( 𝑦 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) = if ( 𝑧 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑧 ) , if ( 𝑧 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) )
220 219 adantl ⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑧 ∈ ℝ ) ∧ ¬ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) ∧ 𝑦 = 𝑧 ) → if ( 𝑦 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑦 ) , if ( 𝑦 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) = if ( 𝑧 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑧 ) , if ( 𝑧 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) )
221 simplr ⊢ ( ( ( ( ( 𝜑 ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑧 ∈ ℝ ) ∧ ¬ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝑧 ∈ ℝ )
222 iffalse ⊢ ( ¬ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) → if ( 𝑧 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑧 ) , if ( 𝑧 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) = if ( 𝑧 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) )
223 222 adantl ⊢ ( ( ( ( ( 𝜑 ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑧 ∈ ℝ ) ∧ ¬ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → if ( 𝑧 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑧 ) , if ( 𝑧 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) = if ( 𝑧 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) )
224 simp-5r ⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑧 ∈ ℝ ) ∧ ¬ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) ∧ 𝑧 < 𝐴 ) → ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 )
225 simp-4r ⊢ ( ( ( ( ( ( 𝜑 ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑧 ∈ ℝ ) ∧ ¬ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) ∧ ¬ 𝑧 < 𝐴 ) → ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 )
226 224 225 ifclda ⊢ ( ( ( ( ( 𝜑 ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑧 ∈ ℝ ) ∧ ¬ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → if ( 𝑧 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ∈ 𝑢 )
227 223 226 eqeltrd ⊢ ( ( ( ( ( 𝜑 ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑧 ∈ ℝ ) ∧ ¬ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → if ( 𝑧 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑧 ) , if ( 𝑧 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) ∈ 𝑢 )
228 216 220 221 227 fvmptd ⊢ ( ( ( ( ( 𝜑 ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑧 ∈ ℝ ) ∧ ¬ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝐺 ‘ 𝑧 ) = if ( 𝑧 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑧 ) , if ( 𝑧 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) )
229 228 223 eqtrd ⊢ ( ( ( ( ( 𝜑 ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑧 ∈ ℝ ) ∧ ¬ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝐺 ‘ 𝑧 ) = if ( 𝑧 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) )
230 229 226 eqeltrd ⊢ ( ( ( ( ( 𝜑 ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑧 ∈ ℝ ) ∧ ¬ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝐺 ‘ 𝑧 ) ∈ 𝑢 )
231 215 230 sylancom ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑧 ∈ 𝑤 ) ∧ ¬ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝐺 ‘ 𝑧 ) ∈ 𝑢 )
232 231 adantl4r ⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑧 ∈ 𝑤 ) ∧ ¬ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝐺 ‘ 𝑧 ) ∈ 𝑢 )
233 207 232 pm2.61dan ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑧 ∈ 𝑤 ) → ( 𝐺 ‘ 𝑧 ) ∈ 𝑢 )
234 173 174 175 233 syl21anc ⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑦 ∈ ( 𝐺 “ 𝑤 ) ) ∧ 𝑧 ∈ 𝑤 ∧ ( 𝐺 ‘ 𝑧 ) = 𝑦 ) → ( 𝐺 ‘ 𝑧 ) ∈ 𝑢 )
235 172 234 eqeltrd ⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑦 ∈ ( 𝐺 “ 𝑤 ) ) ∧ 𝑧 ∈ 𝑤 ∧ ( 𝐺 ‘ 𝑧 ) = 𝑦 ) → 𝑦 ∈ 𝑢 )
236 235 rexlimdv3a ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑦 ∈ ( 𝐺 “ 𝑤 ) ) → ( ∃ 𝑧 ∈ 𝑤 ( 𝐺 ‘ 𝑧 ) = 𝑦 → 𝑦 ∈ 𝑢 ) )
237 169 236 mpd ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑦 ∈ ( 𝐺 “ 𝑤 ) ) → 𝑦 ∈ 𝑢 )
238 237 ex ⊢ ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( 𝑦 ∈ ( 𝐺 “ 𝑤 ) → 𝑦 ∈ 𝑢 ) )
239 238 alrimiv ⊢ ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ∀ 𝑦 ( 𝑦 ∈ ( 𝐺 “ 𝑤 ) → 𝑦 ∈ 𝑢 ) )
240 162 163 164 239 syl21anc ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ∀ 𝑦 ( 𝑦 ∈ ( 𝐺 “ 𝑤 ) → 𝑦 ∈ 𝑢 ) )
241 df-ss ⊢ ( ( 𝐺 “ 𝑤 ) ⊆ 𝑢 ↔ ∀ 𝑦 ( 𝑦 ∈ ( 𝐺 “ 𝑤 ) → 𝑦 ∈ 𝑢 ) )
242 240 241 sylibr ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( 𝐺 “ 𝑤 ) ⊆ 𝑢 )
243 imaundi ⊢ ( 𝐺 “ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) = ( ( 𝐺 “ ( -∞ (,) 𝐴 ) ) ∪ ( 𝐺 “ ( 𝐵 (,) +∞ ) ) )
244 165 a1i ⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( 𝐺 “ ( -∞ (,) 𝐴 ) ) ) → Fun 𝐺 )
245 fvelima ⊢ ( ( Fun 𝐺 ∧ 𝑡 ∈ ( 𝐺 “ ( -∞ (,) 𝐴 ) ) ) → ∃ 𝑧 ∈ ( -∞ (,) 𝐴 ) ( 𝐺 ‘ 𝑧 ) = 𝑡 )
246 244 245 sylancom ⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( 𝐺 “ ( -∞ (,) 𝐴 ) ) ) → ∃ 𝑧 ∈ ( -∞ (,) 𝐴 ) ( 𝐺 ‘ 𝑧 ) = 𝑡 )
247 simp1l ⊢ ( ( ( 𝜑 ∧ 𝑡 ∈ ( 𝐺 “ ( -∞ (,) 𝐴 ) ) ) ∧ 𝑧 ∈ ( -∞ (,) 𝐴 ) ∧ ( 𝐺 ‘ 𝑧 ) = 𝑡 ) → 𝜑 )
248 simp2 ⊢ ( ( ( 𝜑 ∧ 𝑡 ∈ ( 𝐺 “ ( -∞ (,) 𝐴 ) ) ) ∧ 𝑧 ∈ ( -∞ (,) 𝐴 ) ∧ ( 𝐺 ‘ 𝑧 ) = 𝑡 ) → 𝑧 ∈ ( -∞ (,) 𝐴 ) )
249 simp3 ⊢ ( ( ( 𝜑 ∧ 𝑡 ∈ ( 𝐺 “ ( -∞ (,) 𝐴 ) ) ) ∧ 𝑧 ∈ ( -∞ (,) 𝐴 ) ∧ ( 𝐺 ‘ 𝑧 ) = 𝑡 ) → ( 𝐺 ‘ 𝑧 ) = 𝑡 )
250 64 a1i ⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → 𝐺 = ( 𝑦 ∈ ℝ ↦ if ( 𝑦 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑦 ) , if ( 𝑦 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) ) )
251 219 adantl ⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) ∧ 𝑦 = 𝑧 ) → if ( 𝑦 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑦 ) , if ( 𝑦 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) = if ( 𝑧 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑧 ) , if ( 𝑧 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) )
252 elioore ⊢ ( 𝑧 ∈ ( -∞ (,) 𝐴 ) → 𝑧 ∈ ℝ )
253 252 adantl ⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → 𝑧 ∈ ℝ )
254 elioo3g ⊢ ( 𝑧 ∈ ( -∞ (,) 𝐴 ) ↔ ( ( -∞ ∈ ℝ* ∧ 𝐴 ∈ ℝ* ∧ 𝑧 ∈ ℝ* ) ∧ ( -∞ < 𝑧 ∧ 𝑧 < 𝐴 ) ) )
255 254 biimpi ⊢ ( 𝑧 ∈ ( -∞ (,) 𝐴 ) → ( ( -∞ ∈ ℝ* ∧ 𝐴 ∈ ℝ* ∧ 𝑧 ∈ ℝ* ) ∧ ( -∞ < 𝑧 ∧ 𝑧 < 𝐴 ) ) )
256 255 simprrd ⊢ ( 𝑧 ∈ ( -∞ (,) 𝐴 ) → 𝑧 < 𝐴 )
257 256 adantl ⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → 𝑧 < 𝐴 )
258 ltnle ⊢ ( ( 𝑧 ∈ ℝ ∧ 𝐴 ∈ ℝ ) → ( 𝑧 < 𝐴 ↔ ¬ 𝐴 ≤ 𝑧 ) )
259 252 5 258 syl2anr ⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → ( 𝑧 < 𝐴 ↔ ¬ 𝐴 ≤ 𝑧 ) )
260 257 259 mpbid ⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → ¬ 𝐴 ≤ 𝑧 )
261 260 intn3an2d ⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → ¬ ( 𝑧 ∈ ℝ ∧ 𝐴 ≤ 𝑧 ∧ 𝑧 ≤ 𝐵 ) )
262 5 6 jca ⊢ ( 𝜑 → ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) )
263 262 adantr ⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) )
264 elicc2 ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ↔ ( 𝑧 ∈ ℝ ∧ 𝐴 ≤ 𝑧 ∧ 𝑧 ≤ 𝐵 ) ) )
265 263 264 syl ⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → ( 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ↔ ( 𝑧 ∈ ℝ ∧ 𝐴 ≤ 𝑧 ∧ 𝑧 ≤ 𝐵 ) ) )
266 261 265 mtbird ⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → ¬ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) )
267 266 iffalsed ⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → if ( 𝑧 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑧 ) , if ( 𝑧 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) = if ( 𝑧 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) )
268 256 iftrued ⊢ ( 𝑧 ∈ ( -∞ (,) 𝐴 ) → if ( 𝑧 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) = ( 𝐹 ‘ 𝐴 ) )
269 268 adantl ⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → if ( 𝑧 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) = ( 𝐹 ‘ 𝐴 ) )
270 267 269 eqtrd ⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → if ( 𝑧 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑧 ) , if ( 𝑧 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) = ( 𝐹 ‘ 𝐴 ) )
271 127 adantr ⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → ( 𝐹 ‘ 𝐴 ) ∈ 𝑌 )
272 270 271 eqeltrd ⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → if ( 𝑧 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑧 ) , if ( 𝑧 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) ∈ 𝑌 )
273 250 251 253 272 fvmptd ⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → ( 𝐺 ‘ 𝑧 ) = if ( 𝑧 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑧 ) , if ( 𝑧 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) )
274 273 adantr ⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) ∧ ( 𝐺 ‘ 𝑧 ) = 𝑡 ) → ( 𝐺 ‘ 𝑧 ) = if ( 𝑧 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑧 ) , if ( 𝑧 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) )
275 simpr ⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) ∧ ( 𝐺 ‘ 𝑧 ) = 𝑡 ) → ( 𝐺 ‘ 𝑧 ) = 𝑡 )
276 270 adantr ⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) ∧ ( 𝐺 ‘ 𝑧 ) = 𝑡 ) → if ( 𝑧 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑧 ) , if ( 𝑧 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) = ( 𝐹 ‘ 𝐴 ) )
277 274 275 276 3eqtr3d ⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) ∧ ( 𝐺 ‘ 𝑧 ) = 𝑡 ) → 𝑡 = ( 𝐹 ‘ 𝐴 ) )
278 247 248 249 277 syl21anc ⊢ ( ( ( 𝜑 ∧ 𝑡 ∈ ( 𝐺 “ ( -∞ (,) 𝐴 ) ) ) ∧ 𝑧 ∈ ( -∞ (,) 𝐴 ) ∧ ( 𝐺 ‘ 𝑧 ) = 𝑡 ) → 𝑡 = ( 𝐹 ‘ 𝐴 ) )
279 278 rexlimdv3a ⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( 𝐺 “ ( -∞ (,) 𝐴 ) ) ) → ( ∃ 𝑧 ∈ ( -∞ (,) 𝐴 ) ( 𝐺 ‘ 𝑧 ) = 𝑡 → 𝑡 = ( 𝐹 ‘ 𝐴 ) ) )
280 246 279 mpd ⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( 𝐺 “ ( -∞ (,) 𝐴 ) ) ) → 𝑡 = ( 𝐹 ‘ 𝐴 ) )
281 velsn ⊢ ( 𝑡 ∈ { ( 𝐹 ‘ 𝐴 ) } ↔ 𝑡 = ( 𝐹 ‘ 𝐴 ) )
282 280 281 sylibr ⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( 𝐺 “ ( -∞ (,) 𝐴 ) ) ) → 𝑡 ∈ { ( 𝐹 ‘ 𝐴 ) } )
283 282 ex ⊢ ( 𝜑 → ( 𝑡 ∈ ( 𝐺 “ ( -∞ (,) 𝐴 ) ) → 𝑡 ∈ { ( 𝐹 ‘ 𝐴 ) } ) )
284 283 ssrdv ⊢ ( 𝜑 → ( 𝐺 “ ( -∞ (,) 𝐴 ) ) ⊆ { ( 𝐹 ‘ 𝐴 ) } )
285 284 adantr ⊢ ( ( 𝜑 ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) → ( 𝐺 “ ( -∞ (,) 𝐴 ) ) ⊆ { ( 𝐹 ‘ 𝐴 ) } )
286 simpr ⊢ ( ( 𝜑 ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) → ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 )
287 286 snssd ⊢ ( ( 𝜑 ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) → { ( 𝐹 ‘ 𝐴 ) } ⊆ 𝑢 )
288 285 287 sstrd ⊢ ( ( 𝜑 ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) → ( 𝐺 “ ( -∞ (,) 𝐴 ) ) ⊆ 𝑢 )
289 288 adantr ⊢ ( ( ( 𝜑 ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( 𝐺 “ ( -∞ (,) 𝐴 ) ) ⊆ 𝑢 )
290 fvelima ⊢ ( ( Fun 𝐺 ∧ 𝑡 ∈ ( 𝐺 “ ( 𝐵 (,) +∞ ) ) ) → ∃ 𝑧 ∈ ( 𝐵 (,) +∞ ) ( 𝐺 ‘ 𝑧 ) = 𝑡 )
291 166 290 sylan ⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( 𝐺 “ ( 𝐵 (,) +∞ ) ) ) → ∃ 𝑧 ∈ ( 𝐵 (,) +∞ ) ( 𝐺 ‘ 𝑧 ) = 𝑡 )
292 simp1l ⊢ ( ( ( 𝜑 ∧ 𝑡 ∈ ( 𝐺 “ ( 𝐵 (,) +∞ ) ) ) ∧ 𝑧 ∈ ( 𝐵 (,) +∞ ) ∧ ( 𝐺 ‘ 𝑧 ) = 𝑡 ) → 𝜑 )
293 simp2 ⊢ ( ( ( 𝜑 ∧ 𝑡 ∈ ( 𝐺 “ ( 𝐵 (,) +∞ ) ) ) ∧ 𝑧 ∈ ( 𝐵 (,) +∞ ) ∧ ( 𝐺 ‘ 𝑧 ) = 𝑡 ) → 𝑧 ∈ ( 𝐵 (,) +∞ ) )
294 simp3 ⊢ ( ( ( 𝜑 ∧ 𝑡 ∈ ( 𝐺 “ ( 𝐵 (,) +∞ ) ) ) ∧ 𝑧 ∈ ( 𝐵 (,) +∞ ) ∧ ( 𝐺 ‘ 𝑧 ) = 𝑡 ) → ( 𝐺 ‘ 𝑧 ) = 𝑡 )
295 64 a1i ⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝐵 (,) +∞ ) ) → 𝐺 = ( 𝑦 ∈ ℝ ↦ if ( 𝑦 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑦 ) , if ( 𝑦 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) ) )
296 219 adantl ⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ( 𝐵 (,) +∞ ) ) ∧ 𝑦 = 𝑧 ) → if ( 𝑦 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑦 ) , if ( 𝑦 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) = if ( 𝑧 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑧 ) , if ( 𝑧 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) )
297 elioore ⊢ ( 𝑧 ∈ ( 𝐵 (,) +∞ ) → 𝑧 ∈ ℝ )
298 297 adantl ⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝐵 (,) +∞ ) ) → 𝑧 ∈ ℝ )
299 19 ffvelcdmda ⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝐹 ‘ 𝑧 ) ∈ 𝑌 )
300 299 adantlr ⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ( 𝐵 (,) +∞ ) ) ∧ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝐹 ‘ 𝑧 ) ∈ 𝑌 )
301 5 adantr ⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝐵 (,) +∞ ) ) → 𝐴 ∈ ℝ )
302 6 adantr ⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝐵 (,) +∞ ) ) → 𝐵 ∈ ℝ )
303 7 adantr ⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝐵 (,) +∞ ) ) → 𝐴 ≤ 𝐵 )
304 elioo3g ⊢ ( 𝑧 ∈ ( 𝐵 (,) +∞ ) ↔ ( ( 𝐵 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝑧 ∈ ℝ* ) ∧ ( 𝐵 < 𝑧 ∧ 𝑧 < +∞ ) ) )
305 304 biimpi ⊢ ( 𝑧 ∈ ( 𝐵 (,) +∞ ) → ( ( 𝐵 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝑧 ∈ ℝ* ) ∧ ( 𝐵 < 𝑧 ∧ 𝑧 < +∞ ) ) )
306 305 simprld ⊢ ( 𝑧 ∈ ( 𝐵 (,) +∞ ) → 𝐵 < 𝑧 )
307 306 adantl ⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝐵 (,) +∞ ) ) → 𝐵 < 𝑧 )
308 301 302 298 303 307 lelttrd ⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝐵 (,) +∞ ) ) → 𝐴 < 𝑧 )
309 301 298 308 ltnsymd ⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝐵 (,) +∞ ) ) → ¬ 𝑧 < 𝐴 )
310 309 iffalsed ⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝐵 (,) +∞ ) ) → if ( 𝑧 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) = ( 𝐹 ‘ 𝐵 ) )
311 129 adantr ⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝐵 (,) +∞ ) ) → ( 𝐹 ‘ 𝐵 ) ∈ 𝑌 )
312 310 311 eqeltrd ⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝐵 (,) +∞ ) ) → if ( 𝑧 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ∈ 𝑌 )
313 312 adantr ⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ( 𝐵 (,) +∞ ) ) ∧ ¬ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) → if ( 𝑧 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ∈ 𝑌 )
314 300 313 ifclda ⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝐵 (,) +∞ ) ) → if ( 𝑧 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑧 ) , if ( 𝑧 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) ∈ 𝑌 )
315 295 296 298 314 fvmptd ⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝐵 (,) +∞ ) ) → ( 𝐺 ‘ 𝑧 ) = if ( 𝑧 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑧 ) , if ( 𝑧 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) )
316 315 adantr ⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ( 𝐵 (,) +∞ ) ) ∧ ( 𝐺 ‘ 𝑧 ) = 𝑡 ) → ( 𝐺 ‘ 𝑧 ) = if ( 𝑧 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑧 ) , if ( 𝑧 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) )
317 simpr ⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ( 𝐵 (,) +∞ ) ) ∧ ( 𝐺 ‘ 𝑧 ) = 𝑡 ) → ( 𝐺 ‘ 𝑧 ) = 𝑡 )
318 302 298 ltnled ⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝐵 (,) +∞ ) ) → ( 𝐵 < 𝑧 ↔ ¬ 𝑧 ≤ 𝐵 ) )
319 307 318 mpbid ⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝐵 (,) +∞ ) ) → ¬ 𝑧 ≤ 𝐵 )
320 319 intn3an3d ⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝐵 (,) +∞ ) ) → ¬ ( 𝑧 ∈ ℝ ∧ 𝐴 ≤ 𝑧 ∧ 𝑧 ≤ 𝐵 ) )
321 262 adantr ⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝐵 (,) +∞ ) ) → ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) )
322 321 264 syl ⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝐵 (,) +∞ ) ) → ( 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ↔ ( 𝑧 ∈ ℝ ∧ 𝐴 ≤ 𝑧 ∧ 𝑧 ≤ 𝐵 ) ) )
323 320 322 mtbird ⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝐵 (,) +∞ ) ) → ¬ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) )
324 323 iffalsed ⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝐵 (,) +∞ ) ) → if ( 𝑧 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑧 ) , if ( 𝑧 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) = if ( 𝑧 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) )
325 324 310 eqtrd ⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝐵 (,) +∞ ) ) → if ( 𝑧 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑧 ) , if ( 𝑧 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) = ( 𝐹 ‘ 𝐵 ) )
326 325 adantr ⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ( 𝐵 (,) +∞ ) ) ∧ ( 𝐺 ‘ 𝑧 ) = 𝑡 ) → if ( 𝑧 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑧 ) , if ( 𝑧 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) = ( 𝐹 ‘ 𝐵 ) )
327 316 317 326 3eqtr3d ⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ( 𝐵 (,) +∞ ) ) ∧ ( 𝐺 ‘ 𝑧 ) = 𝑡 ) → 𝑡 = ( 𝐹 ‘ 𝐵 ) )
328 292 293 294 327 syl21anc ⊢ ( ( ( 𝜑 ∧ 𝑡 ∈ ( 𝐺 “ ( 𝐵 (,) +∞ ) ) ) ∧ 𝑧 ∈ ( 𝐵 (,) +∞ ) ∧ ( 𝐺 ‘ 𝑧 ) = 𝑡 ) → 𝑡 = ( 𝐹 ‘ 𝐵 ) )
329 328 rexlimdv3a ⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( 𝐺 “ ( 𝐵 (,) +∞ ) ) ) → ( ∃ 𝑧 ∈ ( 𝐵 (,) +∞ ) ( 𝐺 ‘ 𝑧 ) = 𝑡 → 𝑡 = ( 𝐹 ‘ 𝐵 ) ) )
330 291 329 mpd ⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( 𝐺 “ ( 𝐵 (,) +∞ ) ) ) → 𝑡 = ( 𝐹 ‘ 𝐵 ) )
331 velsn ⊢ ( 𝑡 ∈ { ( 𝐹 ‘ 𝐵 ) } ↔ 𝑡 = ( 𝐹 ‘ 𝐵 ) )
332 330 331 sylibr ⊢ ( ( 𝜑 ∧ 𝑡 ∈ ( 𝐺 “ ( 𝐵 (,) +∞ ) ) ) → 𝑡 ∈ { ( 𝐹 ‘ 𝐵 ) } )
333 332 ex ⊢ ( 𝜑 → ( 𝑡 ∈ ( 𝐺 “ ( 𝐵 (,) +∞ ) ) → 𝑡 ∈ { ( 𝐹 ‘ 𝐵 ) } ) )
334 333 ssrdv ⊢ ( 𝜑 → ( 𝐺 “ ( 𝐵 (,) +∞ ) ) ⊆ { ( 𝐹 ‘ 𝐵 ) } )
335 334 adantr ⊢ ( ( 𝜑 ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( 𝐺 “ ( 𝐵 (,) +∞ ) ) ⊆ { ( 𝐹 ‘ 𝐵 ) } )
336 simpr ⊢ ( ( 𝜑 ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 )
337 336 snssd ⊢ ( ( 𝜑 ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → { ( 𝐹 ‘ 𝐵 ) } ⊆ 𝑢 )
338 335 337 sstrd ⊢ ( ( 𝜑 ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( 𝐺 “ ( 𝐵 (,) +∞ ) ) ⊆ 𝑢 )
339 338 adantlr ⊢ ( ( ( 𝜑 ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( 𝐺 “ ( 𝐵 (,) +∞ ) ) ⊆ 𝑢 )
340 289 339 unssd ⊢ ( ( ( 𝜑 ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( ( 𝐺 “ ( -∞ (,) 𝐴 ) ) ∪ ( 𝐺 “ ( 𝐵 (,) +∞ ) ) ) ⊆ 𝑢 )
341 243 340 eqsstrid ⊢ ( ( ( 𝜑 ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( 𝐺 “ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) ⊆ 𝑢 )
342 159 163 164 341 syl21anc ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( 𝐺 “ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) ⊆ 𝑢 )
343 242 342 unssd ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( ( 𝐺 “ 𝑤 ) ∪ ( 𝐺 “ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) ) ⊆ 𝑢 )
344 158 343 eqsstrid ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( 𝐺 “ ( 𝑤 ∪ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) ) ⊆ 𝑢 )
345 eleq2 ⊢ ( 𝑣 = ( 𝑤 ∪ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) → ( 𝑦 ∈ 𝑣 ↔ 𝑦 ∈ ( 𝑤 ∪ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) ) )
346 imaeq2 ⊢ ( 𝑣 = ( 𝑤 ∪ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) → ( 𝐺 “ 𝑣 ) = ( 𝐺 “ ( 𝑤 ∪ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) ) )
347 346 sseq1d ⊢ ( 𝑣 = ( 𝑤 ∪ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) → ( ( 𝐺 “ 𝑣 ) ⊆ 𝑢 ↔ ( 𝐺 “ ( 𝑤 ∪ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) ) ⊆ 𝑢 ) )
348 345 347 anbi12d ⊢ ( 𝑣 = ( 𝑤 ∪ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) → ( ( 𝑦 ∈ 𝑣 ∧ ( 𝐺 “ 𝑣 ) ⊆ 𝑢 ) ↔ ( 𝑦 ∈ ( 𝑤 ∪ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) ∧ ( 𝐺 “ ( 𝑤 ∪ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) ) ⊆ 𝑢 ) ) )
349 348 rspcev ⊢ ( ( ( 𝑤 ∪ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) ∈ 𝐽 ∧ ( 𝑦 ∈ ( 𝑤 ∪ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) ∧ ( 𝐺 “ ( 𝑤 ∪ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) ) ⊆ 𝑢 ) ) → ∃ 𝑣 ∈ 𝐽 ( 𝑦 ∈ 𝑣 ∧ ( 𝐺 “ 𝑣 ) ⊆ 𝑢 ) )
350 103 157 344 349 syl12anc ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ∃ 𝑣 ∈ 𝐽 ( 𝑦 ∈ 𝑣 ∧ ( 𝐺 “ 𝑣 ) ⊆ 𝑢 ) )
351 79 a1i ⊢ ( 𝑤 ∈ 𝐽 → 𝐽 ∈ Top )
352 iooretop ⊢ ( -∞ (,) 𝐵 ) ∈ ( topGen ‘ ran (,) )
353 352 2 eleqtrri ⊢ ( -∞ (,) 𝐵 ) ∈ 𝐽
354 inopn ⊢ ( ( 𝐽 ∈ Top ∧ 𝑤 ∈ 𝐽 ∧ ( -∞ (,) 𝐵 ) ∈ 𝐽 ) → ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∈ 𝐽 )
355 79 353 354 mp3an13 ⊢ ( 𝑤 ∈ 𝐽 → ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∈ 𝐽 )
356 96 a1i ⊢ ( 𝑤 ∈ 𝐽 → ( -∞ (,) 𝐴 ) ∈ 𝐽 )
357 unopn ⊢ ( ( 𝐽 ∈ Top ∧ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∈ 𝐽 ∧ ( -∞ (,) 𝐴 ) ∈ 𝐽 ) → ( ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∪ ( -∞ (,) 𝐴 ) ) ∈ 𝐽 )
358 351 355 356 357 syl3anc ⊢ ( 𝑤 ∈ 𝐽 → ( ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∪ ( -∞ (,) 𝐴 ) ) ∈ 𝐽 )
359 358 3ad2ant2 ⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) → ( ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∪ ( -∞ (,) 𝐴 ) ) ∈ 𝐽 )
360 359 ad2antrr ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∪ ( -∞ (,) 𝐴 ) ) ∈ 𝐽 )
361 simpll1 ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) )
362 simpll3 ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) )
363 simpr ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 )
364 simpll ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐴 ) ) → ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) )
365 262 ad3antrrr ⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) )
366 eqimss ⊢ ( ( ℝ ∖ ( 𝐴 [,] 𝐵 ) ) = ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) → ( ℝ ∖ ( 𝐴 [,] 𝐵 ) ) ⊆ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) )
367 109 366 syl ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( ℝ ∖ ( 𝐴 [,] 𝐵 ) ) ⊆ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) )
368 difcom ⊢ ( ( ℝ ∖ ( 𝐴 [,] 𝐵 ) ) ⊆ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ↔ ( ℝ ∖ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) ⊆ ( 𝐴 [,] 𝐵 ) )
369 367 368 sylib ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( ℝ ∖ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) ⊆ ( 𝐴 [,] 𝐵 ) )
370 365 369 syl ⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( ℝ ∖ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) ⊆ ( 𝐴 [,] 𝐵 ) )
371 370 adantr ⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐴 ) ) → ( ℝ ∖ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) ⊆ ( 𝐴 [,] 𝐵 ) )
372 simp-4r ⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐴 ) ) → 𝑦 ∈ ℝ )
373 simpr ⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐴 ) ) → ¬ 𝑦 ∈ ( -∞ (,) 𝐴 ) )
374 elioore ⊢ ( 𝑦 ∈ ( 𝐵 (,) +∞ ) → 𝑦 ∈ ℝ )
375 374 adantl ⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → 𝑦 ∈ ℝ )
376 elioo3g ⊢ ( 𝑦 ∈ ( 𝐵 (,) +∞ ) ↔ ( ( 𝐵 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝑦 ∈ ℝ* ) ∧ ( 𝐵 < 𝑦 ∧ 𝑦 < +∞ ) ) )
377 376 biimpi ⊢ ( 𝑦 ∈ ( 𝐵 (,) +∞ ) → ( ( 𝐵 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝑦 ∈ ℝ* ) ∧ ( 𝐵 < 𝑦 ∧ 𝑦 < +∞ ) ) )
378 377 simprld ⊢ ( 𝑦 ∈ ( 𝐵 (,) +∞ ) → 𝐵 < 𝑦 )
379 378 adantl ⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → 𝐵 < 𝑦 )
380 6 adantr ⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → 𝐵 ∈ ℝ )
381 380 375 ltnled ⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → ( 𝐵 < 𝑦 ↔ ¬ 𝑦 ≤ 𝐵 ) )
382 379 381 mpbid ⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → ¬ 𝑦 ≤ 𝐵 )
383 382 intn3an3d ⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → ¬ ( 𝑦 ∈ ℝ ∧ 𝐴 ≤ 𝑦 ∧ 𝑦 ≤ 𝐵 ) )
384 262 adantr ⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) )
385 elicc2 ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ↔ ( 𝑦 ∈ ℝ ∧ 𝐴 ≤ 𝑦 ∧ 𝑦 ≤ 𝐵 ) ) )
386 384 385 syl ⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → ( 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ↔ ( 𝑦 ∈ ℝ ∧ 𝐴 ≤ 𝑦 ∧ 𝑦 ≤ 𝐵 ) ) )
387 383 386 mtbird ⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → ¬ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) )
388 387 iffalsed ⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → if ( 𝑦 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑦 ) , if ( 𝑦 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) = if ( 𝑦 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) )
389 5 adantr ⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → 𝐴 ∈ ℝ )
390 7 adantr ⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → 𝐴 ≤ 𝐵 )
391 389 380 375 390 379 lelttrd ⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → 𝐴 < 𝑦 )
392 389 375 391 ltnsymd ⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → ¬ 𝑦 < 𝐴 )
393 392 iffalsed ⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → if ( 𝑦 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) = ( 𝐹 ‘ 𝐵 ) )
394 388 393 eqtrd ⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → if ( 𝑦 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑦 ) , if ( 𝑦 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) = ( 𝐹 ‘ 𝐵 ) )
395 129 adantr ⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → ( 𝐹 ‘ 𝐵 ) ∈ 𝑌 )
396 394 395 eqeltrd ⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → if ( 𝑦 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑦 ) , if ( 𝑦 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) ∈ 𝑌 )
397 375 396 133 syl2anc ⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → ( 𝐺 ‘ 𝑦 ) = if ( 𝑦 ∈ ( 𝐴 [,] 𝐵 ) , ( 𝐹 ‘ 𝑦 ) , if ( 𝑦 < 𝐴 , ( 𝐹 ‘ 𝐴 ) , ( 𝐹 ‘ 𝐵 ) ) ) )
398 397 394 eqtrd ⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → ( 𝐺 ‘ 𝑦 ) = ( 𝐹 ‘ 𝐵 ) )
399 398 eqcomd ⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → ( 𝐹 ‘ 𝐵 ) = ( 𝐺 ‘ 𝑦 ) )
400 399 adantlr ⊢ ( ( ( 𝜑 ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → ( 𝐹 ‘ 𝐵 ) = ( 𝐺 ‘ 𝑦 ) )
401 simplr ⊢ ( ( ( 𝜑 ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 )
402 400 401 eqeltrd ⊢ ( ( ( 𝜑 ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 )
403 402 adantllr ⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 )
404 403 stoic1a ⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ¬ 𝑦 ∈ ( 𝐵 (,) +∞ ) )
405 404 adantr ⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐴 ) ) → ¬ 𝑦 ∈ ( 𝐵 (,) +∞ ) )
406 ioran ⊢ ( ¬ ( 𝑦 ∈ ( -∞ (,) 𝐴 ) ∨ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) ↔ ( ¬ 𝑦 ∈ ( -∞ (,) 𝐴 ) ∧ ¬ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) )
407 373 405 406 sylanbrc ⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐴 ) ) → ¬ ( 𝑦 ∈ ( -∞ (,) 𝐴 ) ∨ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) )
408 elun ⊢ ( 𝑦 ∈ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ↔ ( 𝑦 ∈ ( -∞ (,) 𝐴 ) ∨ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) )
409 407 408 sylnibr ⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐴 ) ) → ¬ 𝑦 ∈ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) )
410 372 409 eldifd ⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐴 ) ) → 𝑦 ∈ ( ℝ ∖ ( ( -∞ (,) 𝐴 ) ∪ ( 𝐵 (,) +∞ ) ) ) )
411 371 410 sseldd ⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐴 ) ) → 𝑦 ∈ ( 𝐴 [,] 𝐵 ) )
412 411 adantllr ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐴 ) ) → 𝑦 ∈ ( 𝐴 [,] 𝐵 ) )
413 simp-4l ⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝜑 )
414 simpllr ⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 )
415 simpr ⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝑦 ∈ ( 𝐴 [,] 𝐵 ) )
416 simpr ⊢ ( ( ( 𝜑 ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝑦 ∈ ( 𝐴 [,] 𝐵 ) )
417 138 adantlr ⊢ ( ( ( 𝜑 ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝐹 ‘ 𝑦 ) = ( 𝐺 ‘ 𝑦 ) )
418 simplr ⊢ ( ( ( 𝜑 ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 )
419 417 418 eqeltrd ⊢ ( ( ( 𝜑 ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝐹 ‘ 𝑦 ) ∈ 𝑢 )
420 20 ad2antrr ⊢ ( ( ( 𝜑 ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝐹 Fn ( 𝐴 [,] 𝐵 ) )
421 420 143 syl ⊢ ( ( ( 𝜑 ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝑦 ∈ ( ◡ 𝐹 “ 𝑢 ) ↔ ( 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ∧ ( 𝐹 ‘ 𝑦 ) ∈ 𝑢 ) ) )
422 416 419 421 mpbir2and ⊢ ( ( ( 𝜑 ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝑦 ∈ ( ◡ 𝐹 “ 𝑢 ) )
423 413 414 415 422 syl21anc ⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝑦 ∈ ( ◡ 𝐹 “ 𝑢 ) )
424 simplr ⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) )
425 423 424 eleqtrd ⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) )
426 elinel1 ⊢ ( 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) → 𝑦 ∈ 𝑤 )
427 425 426 syl ⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝑦 ∈ 𝑤 )
428 364 412 427 syl2anc ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐴 ) ) → 𝑦 ∈ 𝑤 )
429 simp-4l ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐴 ) ) → ( 𝜑 ∧ 𝑦 ∈ ℝ ) )
430 simp-4r ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐴 ) ) → ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 )
431 simplr ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐴 ) ) → ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 )
432 simpl ⊢ ( ( 𝜑 ∧ 𝑦 = 𝐵 ) → 𝜑 )
433 simpr ⊢ ( ( 𝜑 ∧ 𝑦 = 𝐵 ) → 𝑦 = 𝐵 )
434 36 adantr ⊢ ( ( 𝜑 ∧ 𝑦 = 𝐵 ) → 𝐵 ∈ ( 𝐴 [,] 𝐵 ) )
435 433 434 eqeltrd ⊢ ( ( 𝜑 ∧ 𝑦 = 𝐵 ) → 𝑦 ∈ ( 𝐴 [,] 𝐵 ) )
436 432 435 137 syl2anc ⊢ ( ( 𝜑 ∧ 𝑦 = 𝐵 ) → ( 𝐺 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑦 ) )
437 433 fveq2d ⊢ ( ( 𝜑 ∧ 𝑦 = 𝐵 ) → ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 𝐵 ) )
438 436 437 eqtrd ⊢ ( ( 𝜑 ∧ 𝑦 = 𝐵 ) → ( 𝐺 ‘ 𝑦 ) = ( 𝐹 ‘ 𝐵 ) )
439 438 ad4ant14 ⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐵 ) ) ∧ 𝑦 = 𝐵 ) → ( 𝐺 ‘ 𝑦 ) = ( 𝐹 ‘ 𝐵 ) )
440 simplll ⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐵 ) ) ∧ ¬ 𝑦 = 𝐵 ) → 𝜑 )
441 27 adantr ⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → 𝐵 ∈ ℝ* )
442 pnfxr ⊢ +∞ ∈ ℝ*
443 442 a1i ⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → +∞ ∈ ℝ* )
444 rexr ⊢ ( 𝑦 ∈ ℝ → 𝑦 ∈ ℝ* )
445 444 adantl ⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → 𝑦 ∈ ℝ* )
446 441 443 445 3jca ⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → ( 𝐵 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝑦 ∈ ℝ* ) )
447 446 ad2antrr ⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐵 ) ) ∧ ¬ 𝑦 = 𝐵 ) → ( 𝐵 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝑦 ∈ ℝ* ) )
448 mnflt ⊢ ( 𝑦 ∈ ℝ → -∞ < 𝑦 )
449 448 ad2antlr ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐵 ) ) → -∞ < 𝑦 )
450 mnfxr ⊢ -∞ ∈ ℝ*
451 450 a1i ⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → -∞ ∈ ℝ* )
452 451 441 445 3jca ⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → ( -∞ ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝑦 ∈ ℝ* ) )
453 elioo3g ⊢ ( 𝑦 ∈ ( -∞ (,) 𝐵 ) ↔ ( ( -∞ ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝑦 ∈ ℝ* ) ∧ ( -∞ < 𝑦 ∧ 𝑦 < 𝐵 ) ) )
454 453 notbii ⊢ ( ¬ 𝑦 ∈ ( -∞ (,) 𝐵 ) ↔ ¬ ( ( -∞ ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝑦 ∈ ℝ* ) ∧ ( -∞ < 𝑦 ∧ 𝑦 < 𝐵 ) ) )
455 454 biimpi ⊢ ( ¬ 𝑦 ∈ ( -∞ (,) 𝐵 ) → ¬ ( ( -∞ ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝑦 ∈ ℝ* ) ∧ ( -∞ < 𝑦 ∧ 𝑦 < 𝐵 ) ) )
456 455 adantl ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐵 ) ) → ¬ ( ( -∞ ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝑦 ∈ ℝ* ) ∧ ( -∞ < 𝑦 ∧ 𝑦 < 𝐵 ) ) )
457 nan ⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐵 ) ) → ¬ ( ( -∞ ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝑦 ∈ ℝ* ) ∧ ( -∞ < 𝑦 ∧ 𝑦 < 𝐵 ) ) ) ↔ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐵 ) ) ∧ ( -∞ ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝑦 ∈ ℝ* ) ) → ¬ ( -∞ < 𝑦 ∧ 𝑦 < 𝐵 ) ) )
458 456 457 mpbi ⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐵 ) ) ∧ ( -∞ ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝑦 ∈ ℝ* ) ) → ¬ ( -∞ < 𝑦 ∧ 𝑦 < 𝐵 ) )
459 452 458 mpidan ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐵 ) ) → ¬ ( -∞ < 𝑦 ∧ 𝑦 < 𝐵 ) )
460 nan ⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐵 ) ) → ¬ ( -∞ < 𝑦 ∧ 𝑦 < 𝐵 ) ) ↔ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐵 ) ) ∧ -∞ < 𝑦 ) → ¬ 𝑦 < 𝐵 ) )
461 459 460 mpbi ⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐵 ) ) ∧ -∞ < 𝑦 ) → ¬ 𝑦 < 𝐵 )
462 449 461 mpdan ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐵 ) ) → ¬ 𝑦 < 𝐵 )
463 462 anim1i ⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐵 ) ) ∧ ¬ 𝑦 = 𝐵 ) → ( ¬ 𝑦 < 𝐵 ∧ ¬ 𝑦 = 𝐵 ) )
464 pm4.56 ⊢ ( ( ¬ 𝑦 < 𝐵 ∧ ¬ 𝑦 = 𝐵 ) ↔ ¬ ( 𝑦 < 𝐵 ∨ 𝑦 = 𝐵 ) )
465 463 464 sylib ⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐵 ) ) ∧ ¬ 𝑦 = 𝐵 ) → ¬ ( 𝑦 < 𝐵 ∨ 𝑦 = 𝐵 ) )
466 simpr ⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → 𝑦 ∈ ℝ )
467 6 adantr ⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → 𝐵 ∈ ℝ )
468 466 467 jca ⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → ( 𝑦 ∈ ℝ ∧ 𝐵 ∈ ℝ ) )
469 468 ad2antrr ⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐵 ) ) ∧ ¬ 𝑦 = 𝐵 ) → ( 𝑦 ∈ ℝ ∧ 𝐵 ∈ ℝ ) )
470 leloe ⊢ ( ( 𝑦 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝑦 ≤ 𝐵 ↔ ( 𝑦 < 𝐵 ∨ 𝑦 = 𝐵 ) ) )
471 469 470 syl ⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐵 ) ) ∧ ¬ 𝑦 = 𝐵 ) → ( 𝑦 ≤ 𝐵 ↔ ( 𝑦 < 𝐵 ∨ 𝑦 = 𝐵 ) ) )
472 465 471 mtbird ⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐵 ) ) ∧ ¬ 𝑦 = 𝐵 ) → ¬ 𝑦 ≤ 𝐵 )
473 6 anim1i ⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → ( 𝐵 ∈ ℝ ∧ 𝑦 ∈ ℝ ) )
474 473 ad2antrr ⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐵 ) ) ∧ ¬ 𝑦 = 𝐵 ) → ( 𝐵 ∈ ℝ ∧ 𝑦 ∈ ℝ ) )
475 ltnle ⊢ ( ( 𝐵 ∈ ℝ ∧ 𝑦 ∈ ℝ ) → ( 𝐵 < 𝑦 ↔ ¬ 𝑦 ≤ 𝐵 ) )
476 474 475 syl ⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐵 ) ) ∧ ¬ 𝑦 = 𝐵 ) → ( 𝐵 < 𝑦 ↔ ¬ 𝑦 ≤ 𝐵 ) )
477 472 476 mpbird ⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐵 ) ) ∧ ¬ 𝑦 = 𝐵 ) → 𝐵 < 𝑦 )
478 simpllr ⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐵 ) ) ∧ ¬ 𝑦 = 𝐵 ) → 𝑦 ∈ ℝ )
479 478 ltpnfd ⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐵 ) ) ∧ ¬ 𝑦 = 𝐵 ) → 𝑦 < +∞ )
480 477 479 jca ⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐵 ) ) ∧ ¬ 𝑦 = 𝐵 ) → ( 𝐵 < 𝑦 ∧ 𝑦 < +∞ ) )
481 447 480 376 sylanbrc ⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐵 ) ) ∧ ¬ 𝑦 = 𝐵 ) → 𝑦 ∈ ( 𝐵 (,) +∞ ) )
482 440 481 398 syl2anc ⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐵 ) ) ∧ ¬ 𝑦 = 𝐵 ) → ( 𝐺 ‘ 𝑦 ) = ( 𝐹 ‘ 𝐵 ) )
483 439 482 pm2.61dan ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐵 ) ) → ( 𝐺 ‘ 𝑦 ) = ( 𝐹 ‘ 𝐵 ) )
484 483 eqcomd ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐵 ) ) → ( 𝐹 ‘ 𝐵 ) = ( 𝐺 ‘ 𝑦 ) )
485 484 adantlr ⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐵 ) ) → ( 𝐹 ‘ 𝐵 ) = ( 𝐺 ‘ 𝑦 ) )
486 simplr ⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐵 ) ) → ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 )
487 485 486 eqeltrd ⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐵 ) ) → ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 )
488 487 stoic1a ⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ¬ ¬ 𝑦 ∈ ( -∞ (,) 𝐵 ) )
489 488 notnotrd ⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → 𝑦 ∈ ( -∞ (,) 𝐵 ) )
490 429 430 431 489 syl21anc ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐴 ) ) → 𝑦 ∈ ( -∞ (,) 𝐵 ) )
491 428 490 elind ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( -∞ (,) 𝐴 ) ) → 𝑦 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) )
492 491 ex ⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( ¬ 𝑦 ∈ ( -∞ (,) 𝐴 ) → 𝑦 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ) )
493 492 orrd ⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( 𝑦 ∈ ( -∞ (,) 𝐴 ) ∨ 𝑦 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ) )
494 493 orcomd ⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( 𝑦 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∨ 𝑦 ∈ ( -∞ (,) 𝐴 ) ) )
495 elun ⊢ ( 𝑦 ∈ ( ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∪ ( -∞ (,) 𝐴 ) ) ↔ ( 𝑦 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∨ 𝑦 ∈ ( -∞ (,) 𝐴 ) ) )
496 494 495 sylibr ⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → 𝑦 ∈ ( ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∪ ( -∞ (,) 𝐴 ) ) )
497 361 362 363 496 syl21anc ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → 𝑦 ∈ ( ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∪ ( -∞ (,) 𝐴 ) ) )
498 104 simpld ⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) → 𝜑 )
499 498 adantr ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → 𝜑 )
500 simpll2 ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → 𝑤 ∈ 𝐽 )
501 11 500 160 sylancr ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → 𝑤 ⊆ ℝ )
502 499 501 jca ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( 𝜑 ∧ 𝑤 ⊆ ℝ ) )
503 simplr ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 )
504 65 ffnd ⊢ ( 𝜑 → 𝐺 Fn ℝ )
505 504 ad3antrrr ⊢ ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) → 𝐺 Fn ℝ )
506 ssinss1 ⊢ ( 𝑤 ⊆ ℝ → ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ⊆ ℝ )
507 506 ad3antlr ⊢ ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) → ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ⊆ ℝ )
508 ioossre ⊢ ( -∞ (,) 𝐴 ) ⊆ ℝ
509 508 a1i ⊢ ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) → ( -∞ (,) 𝐴 ) ⊆ ℝ )
510 unima ⊢ ( ( 𝐺 Fn ℝ ∧ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ⊆ ℝ ∧ ( -∞ (,) 𝐴 ) ⊆ ℝ ) → ( 𝐺 “ ( ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∪ ( -∞ (,) 𝐴 ) ) ) = ( ( 𝐺 “ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ) ∪ ( 𝐺 “ ( -∞ (,) 𝐴 ) ) ) )
511 505 507 509 510 syl3anc ⊢ ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) → ( 𝐺 “ ( ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∪ ( -∞ (,) 𝐴 ) ) ) = ( ( 𝐺 “ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ) ∪ ( 𝐺 “ ( -∞ (,) 𝐴 ) ) ) )
512 165 a1i ⊢ ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ 𝑦 ∈ ( 𝐺 “ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ) ) → Fun 𝐺 )
513 fvelima ⊢ ( ( Fun 𝐺 ∧ 𝑦 ∈ ( 𝐺 “ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ) ) → ∃ 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ( 𝐺 ‘ 𝑧 ) = 𝑦 )
514 512 513 sylancom ⊢ ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ 𝑦 ∈ ( 𝐺 “ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ) ) → ∃ 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ( 𝐺 ‘ 𝑧 ) = 𝑦 )
515 171 3ad2ant3 ⊢ ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∧ ( 𝐺 ‘ 𝑧 ) = 𝑦 ) → 𝑦 = ( 𝐺 ‘ 𝑧 ) )
516 simp-5l ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ) ∧ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → 𝜑 )
517 simpllr ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ) ∧ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 )
518 simpr ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ) ∧ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → 𝑧 ∈ ( -∞ (,) 𝐴 ) )
519 273 267 269 3eqtrd ⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → ( 𝐺 ‘ 𝑧 ) = ( 𝐹 ‘ 𝐴 ) )
520 519 3adant2 ⊢ ( ( 𝜑 ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ∧ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → ( 𝐺 ‘ 𝑧 ) = ( 𝐹 ‘ 𝐴 ) )
521 simp2 ⊢ ( ( 𝜑 ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ∧ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 )
522 520 521 eqeltrd ⊢ ( ( 𝜑 ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ∧ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → ( 𝐺 ‘ 𝑧 ) ∈ 𝑢 )
523 516 517 518 522 syl3anc ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ) ∧ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → ( 𝐺 ‘ 𝑧 ) ∈ 𝑢 )
524 simplll ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ) ∧ ¬ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) )
525 simp-5l ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ) ∧ ¬ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → 𝜑 )
526 simplr ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ) ∧ ¬ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) )
527 simpr ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ) ∧ ¬ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → ¬ 𝑧 ∈ ( -∞ (,) 𝐴 ) )
528 elinel1 ⊢ ( 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) → 𝑧 ∈ 𝑤 )
529 528 3ad2ant2 ⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∧ ¬ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → 𝑧 ∈ 𝑤 )
530 elinel2 ⊢ ( 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) → 𝑧 ∈ ( -∞ (,) 𝐵 ) )
531 elioore ⊢ ( 𝑧 ∈ ( -∞ (,) 𝐵 ) → 𝑧 ∈ ℝ )
532 530 531 syl ⊢ ( 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) → 𝑧 ∈ ℝ )
533 532 3ad2ant2 ⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∧ ¬ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → 𝑧 ∈ ℝ )
534 26 3ad2ant1 ⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∧ ¬ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → 𝐴 ∈ ℝ* )
535 533 rexrd ⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∧ ¬ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → 𝑧 ∈ ℝ* )
536 mnflt ⊢ ( 𝑧 ∈ ℝ → -∞ < 𝑧 )
537 533 536 syl ⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∧ ¬ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → -∞ < 𝑧 )
538 450 a1i ⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∧ ¬ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → -∞ ∈ ℝ* )
539 538 534 535 3jca ⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∧ ¬ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → ( -∞ ∈ ℝ* ∧ 𝐴 ∈ ℝ* ∧ 𝑧 ∈ ℝ* ) )
540 simp3 ⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∧ ¬ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → ¬ 𝑧 ∈ ( -∞ (,) 𝐴 ) )
541 540 254 sylnib ⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∧ ¬ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → ¬ ( ( -∞ ∈ ℝ* ∧ 𝐴 ∈ ℝ* ∧ 𝑧 ∈ ℝ* ) ∧ ( -∞ < 𝑧 ∧ 𝑧 < 𝐴 ) ) )
542 nan ⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∧ ¬ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → ¬ ( ( -∞ ∈ ℝ* ∧ 𝐴 ∈ ℝ* ∧ 𝑧 ∈ ℝ* ) ∧ ( -∞ < 𝑧 ∧ 𝑧 < 𝐴 ) ) ) ↔ ( ( ( 𝜑 ∧ 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∧ ¬ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) ∧ ( -∞ ∈ ℝ* ∧ 𝐴 ∈ ℝ* ∧ 𝑧 ∈ ℝ* ) ) → ¬ ( -∞ < 𝑧 ∧ 𝑧 < 𝐴 ) ) )
543 541 542 mpbi ⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∧ ¬ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) ∧ ( -∞ ∈ ℝ* ∧ 𝐴 ∈ ℝ* ∧ 𝑧 ∈ ℝ* ) ) → ¬ ( -∞ < 𝑧 ∧ 𝑧 < 𝐴 ) )
544 539 543 mpdan ⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∧ ¬ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → ¬ ( -∞ < 𝑧 ∧ 𝑧 < 𝐴 ) )
545 nan ⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∧ ¬ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → ¬ ( -∞ < 𝑧 ∧ 𝑧 < 𝐴 ) ) ↔ ( ( ( 𝜑 ∧ 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∧ ¬ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) ∧ -∞ < 𝑧 ) → ¬ 𝑧 < 𝐴 ) )
546 544 545 mpbi ⊢ ( ( ( 𝜑 ∧ 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∧ ¬ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) ∧ -∞ < 𝑧 ) → ¬ 𝑧 < 𝐴 )
547 537 546 mpdan ⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∧ ¬ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → ¬ 𝑧 < 𝐴 )
548 534 535 547 xrnltled ⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∧ ¬ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → 𝐴 ≤ 𝑧 )
549 simp1 ⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∧ ¬ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → 𝜑 )
550 530 3ad2ant2 ⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∧ ¬ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → 𝑧 ∈ ( -∞ (,) 𝐵 ) )
551 531 adantl ⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( -∞ (,) 𝐵 ) ) → 𝑧 ∈ ℝ )
552 6 adantr ⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( -∞ (,) 𝐵 ) ) → 𝐵 ∈ ℝ )
553 elioo3g ⊢ ( 𝑧 ∈ ( -∞ (,) 𝐵 ) ↔ ( ( -∞ ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝑧 ∈ ℝ* ) ∧ ( -∞ < 𝑧 ∧ 𝑧 < 𝐵 ) ) )
554 553 biimpi ⊢ ( 𝑧 ∈ ( -∞ (,) 𝐵 ) → ( ( -∞ ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝑧 ∈ ℝ* ) ∧ ( -∞ < 𝑧 ∧ 𝑧 < 𝐵 ) ) )
555 554 simprrd ⊢ ( 𝑧 ∈ ( -∞ (,) 𝐵 ) → 𝑧 < 𝐵 )
556 555 adantl ⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( -∞ (,) 𝐵 ) ) → 𝑧 < 𝐵 )
557 551 552 556 ltled ⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( -∞ (,) 𝐵 ) ) → 𝑧 ≤ 𝐵 )
558 549 550 557 syl2anc ⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∧ ¬ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → 𝑧 ≤ 𝐵 )
559 262 3ad2ant1 ⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∧ ¬ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) )
560 559 264 syl ⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∧ ¬ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → ( 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ↔ ( 𝑧 ∈ ℝ ∧ 𝐴 ≤ 𝑧 ∧ 𝑧 ≤ 𝐵 ) ) )
561 533 548 558 560 mpbir3and ⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∧ ¬ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → 𝑧 ∈ ( 𝐴 [,] 𝐵 ) )
562 529 561 elind ⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∧ ¬ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → 𝑧 ∈ ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) )
563 525 526 527 562 syl3anc ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ) ∧ ¬ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → 𝑧 ∈ ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) )
564 elinel2 ⊢ ( 𝑧 ∈ ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) → 𝑧 ∈ ( 𝐴 [,] 𝐵 ) )
565 564 anim2i ⊢ ( ( 𝜑 ∧ 𝑧 ∈ ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) → ( 𝜑 ∧ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) )
566 565 adantlr ⊢ ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑧 ∈ ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) → ( 𝜑 ∧ 𝑧 ∈ ( 𝐴 [,] 𝐵 ) ) )
567 566 186 syl ⊢ ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑧 ∈ ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) → ( 𝐺 ‘ 𝑧 ) = ( 𝐹 ‘ 𝑧 ) )
568 20 ad2antrr ⊢ ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑧 ∈ ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) → 𝐹 Fn ( 𝐴 [,] 𝐵 ) )
569 simpr ⊢ ( ( ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ∧ 𝑧 ∈ ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) → 𝑧 ∈ ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) )
570 195 adantr ⊢ ( ( ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ∧ 𝑧 ∈ ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) → ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) = ( ◡ 𝐹 “ 𝑢 ) )
571 569 570 eleqtrd ⊢ ( ( ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ∧ 𝑧 ∈ ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) → 𝑧 ∈ ( ◡ 𝐹 “ 𝑢 ) )
572 571 adantll ⊢ ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑧 ∈ ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) → 𝑧 ∈ ( ◡ 𝐹 “ 𝑢 ) )
573 201 simplbda ⊢ ( ( 𝐹 Fn ( 𝐴 [,] 𝐵 ) ∧ 𝑧 ∈ ( ◡ 𝐹 “ 𝑢 ) ) → ( 𝐹 ‘ 𝑧 ) ∈ 𝑢 )
574 568 572 573 syl2anc ⊢ ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑧 ∈ ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) → ( 𝐹 ‘ 𝑧 ) ∈ 𝑢 )
575 567 574 eqeltrd ⊢ ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑧 ∈ ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) → ( 𝐺 ‘ 𝑧 ) ∈ 𝑢 )
576 575 adantllr ⊢ ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑧 ∈ ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) → ( 𝐺 ‘ 𝑧 ) ∈ 𝑢 )
577 524 563 576 syl2anc ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ) ∧ ¬ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → ( 𝐺 ‘ 𝑧 ) ∈ 𝑢 )
578 523 577 pm2.61dan ⊢ ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ) → ( 𝐺 ‘ 𝑧 ) ∈ 𝑢 )
579 578 3adant3 ⊢ ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∧ ( 𝐺 ‘ 𝑧 ) = 𝑦 ) → ( 𝐺 ‘ 𝑧 ) ∈ 𝑢 )
580 515 579 eqeltrd ⊢ ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∧ ( 𝐺 ‘ 𝑧 ) = 𝑦 ) → 𝑦 ∈ 𝑢 )
581 580 3adant1r ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ 𝑦 ∈ ( 𝐺 “ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ) ) ∧ 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∧ ( 𝐺 ‘ 𝑧 ) = 𝑦 ) → 𝑦 ∈ 𝑢 )
582 581 rexlimdv3a ⊢ ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ 𝑦 ∈ ( 𝐺 “ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ) ) → ( ∃ 𝑧 ∈ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ( 𝐺 ‘ 𝑧 ) = 𝑦 → 𝑦 ∈ 𝑢 ) )
583 514 582 mpd ⊢ ( ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ 𝑦 ∈ ( 𝐺 “ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ) ) → 𝑦 ∈ 𝑢 )
584 583 ex ⊢ ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) → ( 𝑦 ∈ ( 𝐺 “ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ) → 𝑦 ∈ 𝑢 ) )
585 584 ssrdv ⊢ ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) → ( 𝐺 “ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ) ⊆ 𝑢 )
586 288 ad4ant14 ⊢ ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) → ( 𝐺 “ ( -∞ (,) 𝐴 ) ) ⊆ 𝑢 )
587 585 586 unssd ⊢ ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) → ( ( 𝐺 “ ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ) ∪ ( 𝐺 “ ( -∞ (,) 𝐴 ) ) ) ⊆ 𝑢 )
588 511 587 eqsstrd ⊢ ( ( ( ( 𝜑 ∧ 𝑤 ⊆ ℝ ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) → ( 𝐺 “ ( ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∪ ( -∞ (,) 𝐴 ) ) ) ⊆ 𝑢 )
589 502 362 503 588 syl21anc ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( 𝐺 “ ( ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∪ ( -∞ (,) 𝐴 ) ) ) ⊆ 𝑢 )
590 eleq2 ⊢ ( 𝑣 = ( ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∪ ( -∞ (,) 𝐴 ) ) → ( 𝑦 ∈ 𝑣 ↔ 𝑦 ∈ ( ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∪ ( -∞ (,) 𝐴 ) ) ) )
591 imaeq2 ⊢ ( 𝑣 = ( ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∪ ( -∞ (,) 𝐴 ) ) → ( 𝐺 “ 𝑣 ) = ( 𝐺 “ ( ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∪ ( -∞ (,) 𝐴 ) ) ) )
592 591 sseq1d ⊢ ( 𝑣 = ( ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∪ ( -∞ (,) 𝐴 ) ) → ( ( 𝐺 “ 𝑣 ) ⊆ 𝑢 ↔ ( 𝐺 “ ( ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∪ ( -∞ (,) 𝐴 ) ) ) ⊆ 𝑢 ) )
593 590 592 anbi12d ⊢ ( 𝑣 = ( ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∪ ( -∞ (,) 𝐴 ) ) → ( ( 𝑦 ∈ 𝑣 ∧ ( 𝐺 “ 𝑣 ) ⊆ 𝑢 ) ↔ ( 𝑦 ∈ ( ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∪ ( -∞ (,) 𝐴 ) ) ∧ ( 𝐺 “ ( ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∪ ( -∞ (,) 𝐴 ) ) ) ⊆ 𝑢 ) ) )
594 593 rspcev ⊢ ( ( ( ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∪ ( -∞ (,) 𝐴 ) ) ∈ 𝐽 ∧ ( 𝑦 ∈ ( ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∪ ( -∞ (,) 𝐴 ) ) ∧ ( 𝐺 “ ( ( 𝑤 ∩ ( -∞ (,) 𝐵 ) ) ∪ ( -∞ (,) 𝐴 ) ) ) ⊆ 𝑢 ) ) → ∃ 𝑣 ∈ 𝐽 ( 𝑦 ∈ 𝑣 ∧ ( 𝐺 “ 𝑣 ) ⊆ 𝑢 ) )
595 360 497 589 594 syl12anc ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ∃ 𝑣 ∈ 𝐽 ( 𝑦 ∈ 𝑣 ∧ ( 𝐺 “ 𝑣 ) ⊆ 𝑢 ) )
596 350 595 pm2.61dan ⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) → ∃ 𝑣 ∈ 𝐽 ( 𝑦 ∈ 𝑣 ∧ ( 𝐺 “ 𝑣 ) ⊆ 𝑢 ) )
597 simpll2 ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → 𝑤 ∈ 𝐽 )
598 iooretop ⊢ ( 𝐴 (,) +∞ ) ∈ ( topGen ‘ ran (,) )
599 598 2 eleqtrri ⊢ ( 𝐴 (,) +∞ ) ∈ 𝐽
600 inopn ⊢ ( ( 𝐽 ∈ Top ∧ 𝑤 ∈ 𝐽 ∧ ( 𝐴 (,) +∞ ) ∈ 𝐽 ) → ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ∈ 𝐽 )
601 79 599 600 mp3an13 ⊢ ( 𝑤 ∈ 𝐽 → ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ∈ 𝐽 )
602 98 a1i ⊢ ( 𝑤 ∈ 𝐽 → ( 𝐵 (,) +∞ ) ∈ 𝐽 )
603 unopn ⊢ ( ( 𝐽 ∈ Top ∧ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ∈ 𝐽 ∧ ( 𝐵 (,) +∞ ) ∈ 𝐽 ) → ( ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ∪ ( 𝐵 (,) +∞ ) ) ∈ 𝐽 )
604 351 601 602 603 syl3anc ⊢ ( 𝑤 ∈ 𝐽 → ( ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ∪ ( 𝐵 (,) +∞ ) ) ∈ 𝐽 )
605 597 604 syl ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ∪ ( 𝐵 (,) +∞ ) ) ∈ 𝐽 )
606 simplll ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) )
607 606 simpld ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → ( 𝜑 ∧ 𝑦 ∈ ℝ ) )
608 607 simpld ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → 𝜑 )
609 simp-4r ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 )
610 simp-5r ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → 𝑦 ∈ ℝ )
611 simplr ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 )
612 simpll ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑦 < 𝐴 ) → 𝜑 )
613 26 adantr ⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → 𝐴 ∈ ℝ* )
614 451 613 445 3jca ⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → ( -∞ ∈ ℝ* ∧ 𝐴 ∈ ℝ* ∧ 𝑦 ∈ ℝ* ) )
615 614 adantr ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑦 < 𝐴 ) → ( -∞ ∈ ℝ* ∧ 𝐴 ∈ ℝ* ∧ 𝑦 ∈ ℝ* ) )
616 448 anim1i ⊢ ( ( 𝑦 ∈ ℝ ∧ 𝑦 < 𝐴 ) → ( -∞ < 𝑦 ∧ 𝑦 < 𝐴 ) )
617 616 adantll ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑦 < 𝐴 ) → ( -∞ < 𝑦 ∧ 𝑦 < 𝐴 ) )
618 elioo3g ⊢ ( 𝑦 ∈ ( -∞ (,) 𝐴 ) ↔ ( ( -∞ ∈ ℝ* ∧ 𝐴 ∈ ℝ* ∧ 𝑦 ∈ ℝ* ) ∧ ( -∞ < 𝑦 ∧ 𝑦 < 𝐴 ) ) )
619 615 617 618 sylanbrc ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑦 < 𝐴 ) → 𝑦 ∈ ( -∞ (,) 𝐴 ) )
620 eleq1 ⊢ ( 𝑧 = 𝑦 → ( 𝑧 ∈ ( -∞ (,) 𝐴 ) ↔ 𝑦 ∈ ( -∞ (,) 𝐴 ) ) )
621 620 anbi2d ⊢ ( 𝑧 = 𝑦 → ( ( 𝜑 ∧ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) ↔ ( 𝜑 ∧ 𝑦 ∈ ( -∞ (,) 𝐴 ) ) ) )
622 fveq2 ⊢ ( 𝑧 = 𝑦 → ( 𝐺 ‘ 𝑧 ) = ( 𝐺 ‘ 𝑦 ) )
623 622 eqeq1d ⊢ ( 𝑧 = 𝑦 → ( ( 𝐺 ‘ 𝑧 ) = ( 𝐹 ‘ 𝐴 ) ↔ ( 𝐺 ‘ 𝑦 ) = ( 𝐹 ‘ 𝐴 ) ) )
624 621 623 imbi12d ⊢ ( 𝑧 = 𝑦 → ( ( ( 𝜑 ∧ 𝑧 ∈ ( -∞ (,) 𝐴 ) ) → ( 𝐺 ‘ 𝑧 ) = ( 𝐹 ‘ 𝐴 ) ) ↔ ( ( 𝜑 ∧ 𝑦 ∈ ( -∞ (,) 𝐴 ) ) → ( 𝐺 ‘ 𝑦 ) = ( 𝐹 ‘ 𝐴 ) ) ) )
625 624 519 chvarvv ⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( -∞ (,) 𝐴 ) ) → ( 𝐺 ‘ 𝑦 ) = ( 𝐹 ‘ 𝐴 ) )
626 612 619 625 syl2anc ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑦 < 𝐴 ) → ( 𝐺 ‘ 𝑦 ) = ( 𝐹 ‘ 𝐴 ) )
627 626 eqcomd ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑦 < 𝐴 ) → ( 𝐹 ‘ 𝐴 ) = ( 𝐺 ‘ 𝑦 ) )
628 627 ad4ant14 ⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ 𝑦 < 𝐴 ) → ( 𝐹 ‘ 𝐴 ) = ( 𝐺 ‘ 𝑦 ) )
629 simpllr ⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ 𝑦 < 𝐴 ) → ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 )
630 628 629 eqeltrd ⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ 𝑦 < 𝐴 ) → ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 )
631 simplr ⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ 𝑦 < 𝐴 ) → ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 )
632 630 631 pm2.65da ⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) → ¬ 𝑦 < 𝐴 )
633 5 anim1i ⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → ( 𝐴 ∈ ℝ ∧ 𝑦 ∈ ℝ ) )
634 633 ad2antrr ⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) → ( 𝐴 ∈ ℝ ∧ 𝑦 ∈ ℝ ) )
635 lenlt ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝑦 ∈ ℝ ) → ( 𝐴 ≤ 𝑦 ↔ ¬ 𝑦 < 𝐴 ) )
636 634 635 syl ⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) → ( 𝐴 ≤ 𝑦 ↔ ¬ 𝑦 < 𝐴 ) )
637 632 636 mpbird ⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) → 𝐴 ≤ 𝑦 )
638 606 611 637 syl2anc ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → 𝐴 ≤ 𝑦 )
639 ltpnf ⊢ ( 𝑦 ∈ ℝ → 𝑦 < +∞ )
640 639 ad2antlr ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → 𝑦 < +∞ )
641 446 adantr ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → ( 𝐵 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝑦 ∈ ℝ* ) )
642 376 notbii ⊢ ( ¬ 𝑦 ∈ ( 𝐵 (,) +∞ ) ↔ ¬ ( ( 𝐵 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝑦 ∈ ℝ* ) ∧ ( 𝐵 < 𝑦 ∧ 𝑦 < +∞ ) ) )
643 642 biimpi ⊢ ( ¬ 𝑦 ∈ ( 𝐵 (,) +∞ ) → ¬ ( ( 𝐵 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝑦 ∈ ℝ* ) ∧ ( 𝐵 < 𝑦 ∧ 𝑦 < +∞ ) ) )
644 643 adantl ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → ¬ ( ( 𝐵 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝑦 ∈ ℝ* ) ∧ ( 𝐵 < 𝑦 ∧ 𝑦 < +∞ ) ) )
645 imnan ⊢ ( ( ( 𝐵 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝑦 ∈ ℝ* ) → ¬ ( 𝐵 < 𝑦 ∧ 𝑦 < +∞ ) ) ↔ ¬ ( ( 𝐵 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝑦 ∈ ℝ* ) ∧ ( 𝐵 < 𝑦 ∧ 𝑦 < +∞ ) ) )
646 644 645 sylibr ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → ( ( 𝐵 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝑦 ∈ ℝ* ) → ¬ ( 𝐵 < 𝑦 ∧ 𝑦 < +∞ ) ) )
647 641 646 mpd ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → ¬ ( 𝐵 < 𝑦 ∧ 𝑦 < +∞ ) )
648 ancom ⊢ ( ( 𝐵 < 𝑦 ∧ 𝑦 < +∞ ) ↔ ( 𝑦 < +∞ ∧ 𝐵 < 𝑦 ) )
649 647 648 sylnib ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → ¬ ( 𝑦 < +∞ ∧ 𝐵 < 𝑦 ) )
650 imnan ⊢ ( ( 𝑦 < +∞ → ¬ 𝐵 < 𝑦 ) ↔ ¬ ( 𝑦 < +∞ ∧ 𝐵 < 𝑦 ) )
651 649 650 sylibr ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → ( 𝑦 < +∞ → ¬ 𝐵 < 𝑦 ) )
652 640 651 mpd ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → ¬ 𝐵 < 𝑦 )
653 468 adantr ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → ( 𝑦 ∈ ℝ ∧ 𝐵 ∈ ℝ ) )
654 lenlt ⊢ ( ( 𝑦 ∈ ℝ ∧ 𝐵 ∈ ℝ ) → ( 𝑦 ≤ 𝐵 ↔ ¬ 𝐵 < 𝑦 ) )
655 653 654 syl ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → ( 𝑦 ≤ 𝐵 ↔ ¬ 𝐵 < 𝑦 ) )
656 652 655 mpbird ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → 𝑦 ≤ 𝐵 )
657 607 656 sylancom ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → 𝑦 ≤ 𝐵 )
658 262 ad5antr ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) )
659 658 385 syl ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → ( 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ↔ ( 𝑦 ∈ ℝ ∧ 𝐴 ≤ 𝑦 ∧ 𝑦 ≤ 𝐵 ) ) )
660 610 638 657 659 mpbir3and ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → 𝑦 ∈ ( 𝐴 [,] 𝐵 ) )
661 608 609 660 422 syl21anc ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → 𝑦 ∈ ( ◡ 𝐹 “ 𝑢 ) )
662 simpllr ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) )
663 661 662 eleqtrd ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) )
664 663 426 syl ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → 𝑦 ∈ 𝑤 )
665 fveq2 ⊢ ( 𝑦 = 𝐴 → ( 𝐺 ‘ 𝑦 ) = ( 𝐺 ‘ 𝐴 ) )
666 29 ancli ⊢ ( 𝜑 → ( 𝜑 ∧ 𝐴 ∈ ( 𝐴 [,] 𝐵 ) ) )
667 eleq1 ⊢ ( 𝑦 = 𝐴 → ( 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ↔ 𝐴 ∈ ( 𝐴 [,] 𝐵 ) ) )
668 667 anbi2d ⊢ ( 𝑦 = 𝐴 → ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) ↔ ( 𝜑 ∧ 𝐴 ∈ ( 𝐴 [,] 𝐵 ) ) ) )
669 fveq2 ⊢ ( 𝑦 = 𝐴 → ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 𝐴 ) )
670 665 669 eqeq12d ⊢ ( 𝑦 = 𝐴 → ( ( 𝐺 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑦 ) ↔ ( 𝐺 ‘ 𝐴 ) = ( 𝐹 ‘ 𝐴 ) ) )
671 668 670 imbi12d ⊢ ( 𝑦 = 𝐴 → ( ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝐺 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑦 ) ) ↔ ( ( 𝜑 ∧ 𝐴 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝐺 ‘ 𝐴 ) = ( 𝐹 ‘ 𝐴 ) ) ) )
672 671 137 vtoclg ⊢ ( 𝐴 ∈ ℝ → ( ( 𝜑 ∧ 𝐴 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝐺 ‘ 𝐴 ) = ( 𝐹 ‘ 𝐴 ) ) )
673 5 666 672 sylc ⊢ ( 𝜑 → ( 𝐺 ‘ 𝐴 ) = ( 𝐹 ‘ 𝐴 ) )
674 665 673 sylan9eqr ⊢ ( ( 𝜑 ∧ 𝑦 = 𝐴 ) → ( 𝐺 ‘ 𝑦 ) = ( 𝐹 ‘ 𝐴 ) )
675 674 ad4ant14 ⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( 𝐴 (,) +∞ ) ) ∧ 𝑦 = 𝐴 ) → ( 𝐺 ‘ 𝑦 ) = ( 𝐹 ‘ 𝐴 ) )
676 simplll ⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( 𝐴 (,) +∞ ) ) ∧ ¬ 𝑦 = 𝐴 ) → 𝜑 )
677 614 ad2antrr ⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( 𝐴 (,) +∞ ) ) ∧ ¬ 𝑦 = 𝐴 ) → ( -∞ ∈ ℝ* ∧ 𝐴 ∈ ℝ* ∧ 𝑦 ∈ ℝ* ) )
678 448 ad3antlr ⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( 𝐴 (,) +∞ ) ) ∧ ¬ 𝑦 = 𝐴 ) → -∞ < 𝑦 )
679 simpllr ⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( 𝐴 (,) +∞ ) ) ∧ ¬ 𝑦 = 𝐴 ) → 𝑦 ∈ ℝ )
680 676 5 syl ⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( 𝐴 (,) +∞ ) ) ∧ ¬ 𝑦 = 𝐴 ) → 𝐴 ∈ ℝ )
681 445 adantr ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( 𝐴 (,) +∞ ) ) → 𝑦 ∈ ℝ* )
682 26 ad2antrr ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( 𝐴 (,) +∞ ) ) → 𝐴 ∈ ℝ* )
683 639 ad2antlr ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( 𝐴 (,) +∞ ) ) → 𝑦 < +∞ )
684 simpr ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( 𝐴 (,) +∞ ) ) → ¬ 𝑦 ∈ ( 𝐴 (,) +∞ ) )
685 442 a1i ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( 𝐴 (,) +∞ ) ) → +∞ ∈ ℝ* )
686 682 685 681 3jca ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( 𝐴 (,) +∞ ) ) → ( 𝐴 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝑦 ∈ ℝ* ) )
687 elioo3g ⊢ ( 𝑦 ∈ ( 𝐴 (,) +∞ ) ↔ ( ( 𝐴 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝑦 ∈ ℝ* ) ∧ ( 𝐴 < 𝑦 ∧ 𝑦 < +∞ ) ) )
688 687 notbii ⊢ ( ¬ 𝑦 ∈ ( 𝐴 (,) +∞ ) ↔ ¬ ( ( 𝐴 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝑦 ∈ ℝ* ) ∧ ( 𝐴 < 𝑦 ∧ 𝑦 < +∞ ) ) )
689 688 biimpi ⊢ ( ¬ 𝑦 ∈ ( 𝐴 (,) +∞ ) → ¬ ( ( 𝐴 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝑦 ∈ ℝ* ) ∧ ( 𝐴 < 𝑦 ∧ 𝑦 < +∞ ) ) )
690 nan ⊢ ( ( ¬ 𝑦 ∈ ( 𝐴 (,) +∞ ) → ¬ ( ( 𝐴 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝑦 ∈ ℝ* ) ∧ ( 𝐴 < 𝑦 ∧ 𝑦 < +∞ ) ) ) ↔ ( ( ¬ 𝑦 ∈ ( 𝐴 (,) +∞ ) ∧ ( 𝐴 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝑦 ∈ ℝ* ) ) → ¬ ( 𝐴 < 𝑦 ∧ 𝑦 < +∞ ) ) )
691 689 690 mpbi ⊢ ( ( ¬ 𝑦 ∈ ( 𝐴 (,) +∞ ) ∧ ( 𝐴 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝑦 ∈ ℝ* ) ) → ¬ ( 𝐴 < 𝑦 ∧ 𝑦 < +∞ ) )
692 684 686 691 syl2anc ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( 𝐴 (,) +∞ ) ) → ¬ ( 𝐴 < 𝑦 ∧ 𝑦 < +∞ ) )
693 ancom ⊢ ( ( 𝐴 < 𝑦 ∧ 𝑦 < +∞ ) ↔ ( 𝑦 < +∞ ∧ 𝐴 < 𝑦 ) )
694 692 693 sylnib ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( 𝐴 (,) +∞ ) ) → ¬ ( 𝑦 < +∞ ∧ 𝐴 < 𝑦 ) )
695 nan ⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( 𝐴 (,) +∞ ) ) → ¬ ( 𝑦 < +∞ ∧ 𝐴 < 𝑦 ) ) ↔ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( 𝐴 (,) +∞ ) ) ∧ 𝑦 < +∞ ) → ¬ 𝐴 < 𝑦 ) )
696 694 695 mpbi ⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( 𝐴 (,) +∞ ) ) ∧ 𝑦 < +∞ ) → ¬ 𝐴 < 𝑦 )
697 683 696 mpdan ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( 𝐴 (,) +∞ ) ) → ¬ 𝐴 < 𝑦 )
698 681 682 697 xrnltled ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( 𝐴 (,) +∞ ) ) → 𝑦 ≤ 𝐴 )
699 698 adantr ⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( 𝐴 (,) +∞ ) ) ∧ ¬ 𝑦 = 𝐴 ) → 𝑦 ≤ 𝐴 )
700 neqne ⊢ ( ¬ 𝑦 = 𝐴 → 𝑦 ≠ 𝐴 )
701 700 necomd ⊢ ( ¬ 𝑦 = 𝐴 → 𝐴 ≠ 𝑦 )
702 701 adantl ⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( 𝐴 (,) +∞ ) ) ∧ ¬ 𝑦 = 𝐴 ) → 𝐴 ≠ 𝑦 )
703 679 680 699 702 leneltd ⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( 𝐴 (,) +∞ ) ) ∧ ¬ 𝑦 = 𝐴 ) → 𝑦 < 𝐴 )
704 678 703 jca ⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( 𝐴 (,) +∞ ) ) ∧ ¬ 𝑦 = 𝐴 ) → ( -∞ < 𝑦 ∧ 𝑦 < 𝐴 ) )
705 677 704 618 sylanbrc ⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( 𝐴 (,) +∞ ) ) ∧ ¬ 𝑦 = 𝐴 ) → 𝑦 ∈ ( -∞ (,) 𝐴 ) )
706 676 705 625 syl2anc ⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( 𝐴 (,) +∞ ) ) ∧ ¬ 𝑦 = 𝐴 ) → ( 𝐺 ‘ 𝑦 ) = ( 𝐹 ‘ 𝐴 ) )
707 675 706 pm2.61dan ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( 𝐴 (,) +∞ ) ) → ( 𝐺 ‘ 𝑦 ) = ( 𝐹 ‘ 𝐴 ) )
708 707 eqcomd ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ¬ 𝑦 ∈ ( 𝐴 (,) +∞ ) ) → ( 𝐹 ‘ 𝐴 ) = ( 𝐺 ‘ 𝑦 ) )
709 708 ad4ant14 ⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( 𝐴 (,) +∞ ) ) → ( 𝐹 ‘ 𝐴 ) = ( 𝐺 ‘ 𝑦 ) )
710 simpllr ⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( 𝐴 (,) +∞ ) ) → ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 )
711 709 710 eqeltrd ⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( 𝐴 (,) +∞ ) ) → ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 )
712 simplr ⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( 𝐴 (,) +∞ ) ) → ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 )
713 711 712 condan ⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) → 𝑦 ∈ ( 𝐴 (,) +∞ ) )
714 606 611 713 syl2anc ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → 𝑦 ∈ ( 𝐴 (,) +∞ ) )
715 664 714 elind ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) )
716 715 adantlr ⊢ ( ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) )
717 pm5.6 ⊢ ( ( ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ ¬ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) → 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) ↔ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( 𝑦 ∈ ( 𝐵 (,) +∞ ) ∨ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) ) )
718 716 717 mpbi ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( 𝑦 ∈ ( 𝐵 (,) +∞ ) ∨ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) )
719 718 orcomd ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ∨ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) )
720 elun ⊢ ( 𝑦 ∈ ( ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ∪ ( 𝐵 (,) +∞ ) ) ↔ ( 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ∨ 𝑦 ∈ ( 𝐵 (,) +∞ ) ) )
721 719 720 sylibr ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → 𝑦 ∈ ( ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ∪ ( 𝐵 (,) +∞ ) ) )
722 721 3adantll2 ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → 𝑦 ∈ ( ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ∪ ( 𝐵 (,) +∞ ) ) )
723 simp1ll ⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) → 𝜑 )
724 723 ad2antrr ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → 𝜑 )
725 simpll3 ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) )
726 simpr ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 )
727 504 ad2antrr ⊢ ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → 𝐺 Fn ℝ )
728 ioossre ⊢ ( 𝐴 (,) +∞ ) ⊆ ℝ
729 728 olci ⊢ ( 𝑤 ⊆ ℝ ∨ ( 𝐴 (,) +∞ ) ⊆ ℝ )
730 inss ⊢ ( ( 𝑤 ⊆ ℝ ∨ ( 𝐴 (,) +∞ ) ⊆ ℝ ) → ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ⊆ ℝ )
731 729 730 ax-mp ⊢ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ⊆ ℝ
732 731 a1i ⊢ ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ⊆ ℝ )
733 ioossre ⊢ ( 𝐵 (,) +∞ ) ⊆ ℝ
734 733 a1i ⊢ ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( 𝐵 (,) +∞ ) ⊆ ℝ )
735 unima ⊢ ( ( 𝐺 Fn ℝ ∧ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ⊆ ℝ ∧ ( 𝐵 (,) +∞ ) ⊆ ℝ ) → ( 𝐺 “ ( ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ∪ ( 𝐵 (,) +∞ ) ) ) = ( ( 𝐺 “ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) ∪ ( 𝐺 “ ( 𝐵 (,) +∞ ) ) ) )
736 727 732 734 735 syl3anc ⊢ ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( 𝐺 “ ( ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ∪ ( 𝐵 (,) +∞ ) ) ) = ( ( 𝐺 “ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) ∪ ( 𝐺 “ ( 𝐵 (,) +∞ ) ) ) )
737 simpll ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) ∧ 𝐵 < 𝑦 ) → 𝜑 )
738 731 sseli ⊢ ( 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) → 𝑦 ∈ ℝ )
739 738 ad2antlr ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) ∧ 𝐵 < 𝑦 ) → 𝑦 ∈ ℝ )
740 737 739 446 syl2anc ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) ∧ 𝐵 < 𝑦 ) → ( 𝐵 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝑦 ∈ ℝ* ) )
741 simpr ⊢ ( ( 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ∧ 𝐵 < 𝑦 ) → 𝐵 < 𝑦 )
742 738 ltpnfd ⊢ ( 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) → 𝑦 < +∞ )
743 742 adantr ⊢ ( ( 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ∧ 𝐵 < 𝑦 ) → 𝑦 < +∞ )
744 741 743 jca ⊢ ( ( 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ∧ 𝐵 < 𝑦 ) → ( 𝐵 < 𝑦 ∧ 𝑦 < +∞ ) )
745 744 adantll ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) ∧ 𝐵 < 𝑦 ) → ( 𝐵 < 𝑦 ∧ 𝑦 < +∞ ) )
746 740 745 376 sylanbrc ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) ∧ 𝐵 < 𝑦 ) → 𝑦 ∈ ( 𝐵 (,) +∞ ) )
747 737 746 398 syl2anc ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) ∧ 𝐵 < 𝑦 ) → ( 𝐺 ‘ 𝑦 ) = ( 𝐹 ‘ 𝐵 ) )
748 747 adantllr ⊢ ( ( ( ( 𝜑 ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) ∧ 𝐵 < 𝑦 ) → ( 𝐺 ‘ 𝑦 ) = ( 𝐹 ‘ 𝐵 ) )
749 simpllr ⊢ ( ( ( ( 𝜑 ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) ∧ 𝐵 < 𝑦 ) → ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 )
750 748 749 eqeltrd ⊢ ( ( ( ( 𝜑 ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) ∧ 𝐵 < 𝑦 ) → ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 )
751 750 adantl3r ⊢ ( ( ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) ∧ 𝐵 < 𝑦 ) → ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 )
752 simp-4l ⊢ ( ( ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) ∧ ¬ 𝐵 < 𝑦 ) → 𝜑 )
753 simplr ⊢ ( ( ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) ∧ ¬ 𝐵 < 𝑦 ) → 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) )
754 simpr ⊢ ( ( ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) ∧ ¬ 𝐵 < 𝑦 ) → ¬ 𝐵 < 𝑦 )
755 simpll ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) ∧ ¬ 𝐵 < 𝑦 ) → 𝜑 )
756 738 adantl ⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) → 𝑦 ∈ ℝ )
757 756 adantr ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) ∧ ¬ 𝐵 < 𝑦 ) → 𝑦 ∈ ℝ )
758 5 adantr ⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) → 𝐴 ∈ ℝ )
759 elinel2 ⊢ ( 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) → 𝑦 ∈ ( 𝐴 (,) +∞ ) )
760 687 biimpi ⊢ ( 𝑦 ∈ ( 𝐴 (,) +∞ ) → ( ( 𝐴 ∈ ℝ* ∧ +∞ ∈ ℝ* ∧ 𝑦 ∈ ℝ* ) ∧ ( 𝐴 < 𝑦 ∧ 𝑦 < +∞ ) ) )
761 760 simprld ⊢ ( 𝑦 ∈ ( 𝐴 (,) +∞ ) → 𝐴 < 𝑦 )
762 759 761 syl ⊢ ( 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) → 𝐴 < 𝑦 )
763 762 adantl ⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) → 𝐴 < 𝑦 )
764 758 756 763 ltled ⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) → 𝐴 ≤ 𝑦 )
765 764 adantr ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) ∧ ¬ 𝐵 < 𝑦 ) → 𝐴 ≤ 𝑦 )
766 simpr ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) ∧ ¬ 𝐵 < 𝑦 ) → ¬ 𝐵 < 𝑦 )
767 755 757 468 syl2anc ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) ∧ ¬ 𝐵 < 𝑦 ) → ( 𝑦 ∈ ℝ ∧ 𝐵 ∈ ℝ ) )
768 767 654 syl ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) ∧ ¬ 𝐵 < 𝑦 ) → ( 𝑦 ≤ 𝐵 ↔ ¬ 𝐵 < 𝑦 ) )
769 766 768 mpbird ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) ∧ ¬ 𝐵 < 𝑦 ) → 𝑦 ≤ 𝐵 )
770 262 ad2antrr ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) ∧ ¬ 𝐵 < 𝑦 ) → ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) )
771 770 385 syl ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) ∧ ¬ 𝐵 < 𝑦 ) → ( 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ↔ ( 𝑦 ∈ ℝ ∧ 𝐴 ≤ 𝑦 ∧ 𝑦 ≤ 𝐵 ) ) )
772 757 765 769 771 mpbir3and ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) ∧ ¬ 𝐵 < 𝑦 ) → 𝑦 ∈ ( 𝐴 [,] 𝐵 ) )
773 755 772 137 syl2anc ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) ∧ ¬ 𝐵 < 𝑦 ) → ( 𝐺 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑦 ) )
774 752 753 754 773 syl21anc ⊢ ( ( ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) ∧ ¬ 𝐵 < 𝑦 ) → ( 𝐺 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑦 ) )
775 elinel1 ⊢ ( 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) → 𝑦 ∈ 𝑤 )
776 775 ad2antlr ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) ∧ ¬ 𝐵 < 𝑦 ) → 𝑦 ∈ 𝑤 )
777 776 772 jca ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) ∧ ¬ 𝐵 < 𝑦 ) → ( 𝑦 ∈ 𝑤 ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) )
778 777 adantllr ⊢ ( ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) ∧ ¬ 𝐵 < 𝑦 ) → ( 𝑦 ∈ 𝑤 ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) )
779 778 149 sylibr ⊢ ( ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) ∧ ¬ 𝐵 < 𝑦 ) → 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) )
780 195 ad3antlr ⊢ ( ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) ∧ ¬ 𝐵 < 𝑦 ) → ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) = ( ◡ 𝐹 “ 𝑢 ) )
781 779 780 eleqtrd ⊢ ( ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) ∧ ¬ 𝐵 < 𝑦 ) → 𝑦 ∈ ( ◡ 𝐹 “ 𝑢 ) )
782 20 ad3antrrr ⊢ ( ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) ∧ ¬ 𝐵 < 𝑦 ) → 𝐹 Fn ( 𝐴 [,] 𝐵 ) )
783 782 143 syl ⊢ ( ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) ∧ ¬ 𝐵 < 𝑦 ) → ( 𝑦 ∈ ( ◡ 𝐹 “ 𝑢 ) ↔ ( 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ∧ ( 𝐹 ‘ 𝑦 ) ∈ 𝑢 ) ) )
784 781 783 mpbid ⊢ ( ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) ∧ ¬ 𝐵 < 𝑦 ) → ( 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ∧ ( 𝐹 ‘ 𝑦 ) ∈ 𝑢 ) )
785 784 simprd ⊢ ( ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) ∧ ¬ 𝐵 < 𝑦 ) → ( 𝐹 ‘ 𝑦 ) ∈ 𝑢 )
786 785 adantllr ⊢ ( ( ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) ∧ ¬ 𝐵 < 𝑦 ) → ( 𝐹 ‘ 𝑦 ) ∈ 𝑢 )
787 774 786 eqeltrd ⊢ ( ( ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) ∧ ¬ 𝐵 < 𝑦 ) → ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 )
788 751 787 pm2.61dan ⊢ ( ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) → ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 )
789 788 ralrimiva ⊢ ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ∀ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 )
790 504 fndmd ⊢ ( 𝜑 → dom 𝐺 = ℝ )
791 731 790 sseqtrrid ⊢ ( 𝜑 → ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ⊆ dom 𝐺 )
792 166 791 jca ⊢ ( 𝜑 → ( Fun 𝐺 ∧ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ⊆ dom 𝐺 ) )
793 792 ad2antrr ⊢ ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( Fun 𝐺 ∧ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ⊆ dom 𝐺 ) )
794 funimass4 ⊢ ( ( Fun 𝐺 ∧ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ⊆ dom 𝐺 ) → ( ( 𝐺 “ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) ⊆ 𝑢 ↔ ∀ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) )
795 793 794 syl ⊢ ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( ( 𝐺 “ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) ⊆ 𝑢 ↔ ∀ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) )
796 789 795 mpbird ⊢ ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( 𝐺 “ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) ⊆ 𝑢 )
797 338 adantlr ⊢ ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( 𝐺 “ ( 𝐵 (,) +∞ ) ) ⊆ 𝑢 )
798 796 797 unssd ⊢ ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( ( 𝐺 “ ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ) ∪ ( 𝐺 “ ( 𝐵 (,) +∞ ) ) ) ⊆ 𝑢 )
799 736 798 eqsstrd ⊢ ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( 𝐺 “ ( ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ∪ ( 𝐵 (,) +∞ ) ) ) ⊆ 𝑢 )
800 724 725 726 799 syl21anc ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( 𝐺 “ ( ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ∪ ( 𝐵 (,) +∞ ) ) ) ⊆ 𝑢 )
801 eleq2 ⊢ ( 𝑣 = ( ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ∪ ( 𝐵 (,) +∞ ) ) → ( 𝑦 ∈ 𝑣 ↔ 𝑦 ∈ ( ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ∪ ( 𝐵 (,) +∞ ) ) ) )
802 imaeq2 ⊢ ( 𝑣 = ( ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ∪ ( 𝐵 (,) +∞ ) ) → ( 𝐺 “ 𝑣 ) = ( 𝐺 “ ( ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ∪ ( 𝐵 (,) +∞ ) ) ) )
803 802 sseq1d ⊢ ( 𝑣 = ( ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ∪ ( 𝐵 (,) +∞ ) ) → ( ( 𝐺 “ 𝑣 ) ⊆ 𝑢 ↔ ( 𝐺 “ ( ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ∪ ( 𝐵 (,) +∞ ) ) ) ⊆ 𝑢 ) )
804 801 803 anbi12d ⊢ ( 𝑣 = ( ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ∪ ( 𝐵 (,) +∞ ) ) → ( ( 𝑦 ∈ 𝑣 ∧ ( 𝐺 “ 𝑣 ) ⊆ 𝑢 ) ↔ ( 𝑦 ∈ ( ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ∪ ( 𝐵 (,) +∞ ) ) ∧ ( 𝐺 “ ( ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ∪ ( 𝐵 (,) +∞ ) ) ) ⊆ 𝑢 ) ) )
805 804 rspcev ⊢ ( ( ( ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ∪ ( 𝐵 (,) +∞ ) ) ∈ 𝐽 ∧ ( 𝑦 ∈ ( ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ∪ ( 𝐵 (,) +∞ ) ) ∧ ( 𝐺 “ ( ( 𝑤 ∩ ( 𝐴 (,) +∞ ) ) ∪ ( 𝐵 (,) +∞ ) ) ) ⊆ 𝑢 ) ) → ∃ 𝑣 ∈ 𝐽 ( 𝑦 ∈ 𝑣 ∧ ( 𝐺 “ 𝑣 ) ⊆ 𝑢 ) )
806 605 722 800 805 syl12anc ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ∃ 𝑣 ∈ 𝐽 ( 𝑦 ∈ 𝑣 ∧ ( 𝐺 “ 𝑣 ) ⊆ 𝑢 ) )
807 simpll2 ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → 𝑤 ∈ 𝐽 )
808 iooretop ⊢ ( 𝐴 (,) 𝐵 ) ∈ ( topGen ‘ ran (,) )
809 808 2 eleqtrri ⊢ ( 𝐴 (,) 𝐵 ) ∈ 𝐽
810 inopn ⊢ ( ( 𝐽 ∈ Top ∧ 𝑤 ∈ 𝐽 ∧ ( 𝐴 (,) 𝐵 ) ∈ 𝐽 ) → ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ∈ 𝐽 )
811 79 809 810 mp3an13 ⊢ ( 𝑤 ∈ 𝐽 → ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ∈ 𝐽 )
812 807 811 syl ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ∈ 𝐽 )
813 simp-4r ⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → 𝑦 ∈ ℝ )
814 637 adantr ⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → 𝐴 ≤ 𝑦 )
815 simpll ⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( 𝜑 ∧ 𝑦 ∈ ℝ ) )
816 815 404 656 syl2anc ⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → 𝑦 ≤ 𝐵 )
817 816 adantlr ⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → 𝑦 ≤ 𝐵 )
818 simp-4l ⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → 𝜑 )
819 818 262 syl ⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) )
820 819 385 syl ⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ↔ ( 𝑦 ∈ ℝ ∧ 𝐴 ≤ 𝑦 ∧ 𝑦 ≤ 𝐵 ) ) )
821 813 814 817 820 mpbir3and ⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → 𝑦 ∈ ( 𝐴 [,] 𝐵 ) )
822 821 adantllr ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → 𝑦 ∈ ( 𝐴 [,] 𝐵 ) )
823 818 821 137 syl2anc ⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( 𝐺 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑦 ) )
824 823 adantllr ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( 𝐺 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑦 ) )
825 simp-4r ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 )
826 824 825 eqeltrrd ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( 𝐹 ‘ 𝑦 ) ∈ 𝑢 )
827 simp-5l ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → 𝜑 )
828 827 20 syl ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → 𝐹 Fn ( 𝐴 [,] 𝐵 ) )
829 828 143 syl ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( 𝑦 ∈ ( ◡ 𝐹 “ 𝑢 ) ↔ ( 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ∧ ( 𝐹 ‘ 𝑦 ) ∈ 𝑢 ) ) )
830 822 826 829 mpbir2and ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → 𝑦 ∈ ( ◡ 𝐹 “ 𝑢 ) )
831 simpllr ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) )
832 830 831 eleqtrd ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) )
833 832 426 syl ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → 𝑦 ∈ 𝑤 )
834 simp-5r ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → 𝑦 ∈ ℝ )
835 827 834 822 jca31 ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) )
836 simplr ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 )
837 826 836 jca ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( ( 𝐹 ‘ 𝑦 ) ∈ 𝑢 ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) )
838 nelneq ⊢ ( ( ( 𝐹 ‘ 𝑦 ) ∈ 𝑢 ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) → ¬ ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 𝐴 ) )
839 669 necon3bi ⊢ ( ¬ ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 𝐴 ) → 𝑦 ≠ 𝐴 )
840 837 838 839 3syl ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → 𝑦 ≠ 𝐴 )
841 simpr ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 )
842 826 841 jca ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( ( 𝐹 ‘ 𝑦 ) ∈ 𝑢 ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) )
843 nelneq ⊢ ( ( ( 𝐹 ‘ 𝑦 ) ∈ 𝑢 ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ¬ ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 𝐵 ) )
844 fveq2 ⊢ ( 𝑦 = 𝐵 → ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 𝐵 ) )
845 844 necon3bi ⊢ ( ¬ ( 𝐹 ‘ 𝑦 ) = ( 𝐹 ‘ 𝐵 ) → 𝑦 ≠ 𝐵 )
846 842 843 845 3syl ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → 𝑦 ≠ 𝐵 )
847 613 ad3antrrr ⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) ∧ 𝑦 ≠ 𝐴 ) ∧ 𝑦 ≠ 𝐵 ) → 𝐴 ∈ ℝ* )
848 441 ad3antrrr ⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) ∧ 𝑦 ≠ 𝐴 ) ∧ 𝑦 ≠ 𝐵 ) → 𝐵 ∈ ℝ* )
849 444 ad4antlr ⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) ∧ 𝑦 ≠ 𝐴 ) ∧ 𝑦 ≠ 𝐵 ) → 𝑦 ∈ ℝ* )
850 847 848 849 3jca ⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) ∧ 𝑦 ≠ 𝐴 ) ∧ 𝑦 ≠ 𝐵 ) → ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝑦 ∈ ℝ* ) )
851 simpr ⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) ∧ 𝑦 ≠ 𝐴 ) → 𝑦 ≠ 𝐴 )
852 5 ad2antrr ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝐴 ∈ ℝ )
853 simplr ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝑦 ∈ ℝ )
854 262 adantr ⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ) )
855 854 385 syl ⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ↔ ( 𝑦 ∈ ℝ ∧ 𝐴 ≤ 𝑦 ∧ 𝑦 ≤ 𝐵 ) ) )
856 135 855 mpbid ⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝑦 ∈ ℝ ∧ 𝐴 ≤ 𝑦 ∧ 𝑦 ≤ 𝐵 ) )
857 856 simp2d ⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝐴 ≤ 𝑦 )
858 857 adantlr ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝐴 ≤ 𝑦 )
859 852 853 858 3jca ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝐴 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝐴 ≤ 𝑦 ) )
860 859 adantr ⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) ∧ 𝑦 ≠ 𝐴 ) → ( 𝐴 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝐴 ≤ 𝑦 ) )
861 leltne ⊢ ( ( 𝐴 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝐴 ≤ 𝑦 ) → ( 𝐴 < 𝑦 ↔ 𝑦 ≠ 𝐴 ) )
862 860 861 syl ⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) ∧ 𝑦 ≠ 𝐴 ) → ( 𝐴 < 𝑦 ↔ 𝑦 ≠ 𝐴 ) )
863 851 862 mpbird ⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) ∧ 𝑦 ≠ 𝐴 ) → 𝐴 < 𝑦 )
864 863 adantr ⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) ∧ 𝑦 ≠ 𝐴 ) ∧ 𝑦 ≠ 𝐵 ) → 𝐴 < 𝑦 )
865 necom ⊢ ( 𝑦 ≠ 𝐵 ↔ 𝐵 ≠ 𝑦 )
866 865 biimpi ⊢ ( 𝑦 ≠ 𝐵 → 𝐵 ≠ 𝑦 )
867 866 adantl ⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) ∧ 𝑦 ≠ 𝐵 ) → 𝐵 ≠ 𝑦 )
868 6 ad2antrr ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝐵 ∈ ℝ )
869 856 simp3d ⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝑦 ≤ 𝐵 )
870 869 adantlr ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → 𝑦 ≤ 𝐵 )
871 853 868 870 3jca ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → ( 𝑦 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑦 ≤ 𝐵 ) )
872 871 adantr ⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) ∧ 𝑦 ≠ 𝐵 ) → ( 𝑦 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑦 ≤ 𝐵 ) )
873 leltne ⊢ ( ( 𝑦 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ 𝑦 ≤ 𝐵 ) → ( 𝑦 < 𝐵 ↔ 𝐵 ≠ 𝑦 ) )
874 872 873 syl ⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) ∧ 𝑦 ≠ 𝐵 ) → ( 𝑦 < 𝐵 ↔ 𝐵 ≠ 𝑦 ) )
875 867 874 mpbird ⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) ∧ 𝑦 ≠ 𝐵 ) → 𝑦 < 𝐵 )
876 875 adantlr ⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) ∧ 𝑦 ≠ 𝐴 ) ∧ 𝑦 ≠ 𝐵 ) → 𝑦 < 𝐵 )
877 864 876 jca ⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) ∧ 𝑦 ≠ 𝐴 ) ∧ 𝑦 ≠ 𝐵 ) → ( 𝐴 < 𝑦 ∧ 𝑦 < 𝐵 ) )
878 elioo3g ⊢ ( 𝑦 ∈ ( 𝐴 (,) 𝐵 ) ↔ ( ( 𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ* ∧ 𝑦 ∈ ℝ* ) ∧ ( 𝐴 < 𝑦 ∧ 𝑦 < 𝐵 ) ) )
879 850 877 878 sylanbrc ⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) ∧ 𝑦 ≠ 𝐴 ) ∧ 𝑦 ≠ 𝐵 ) → 𝑦 ∈ ( 𝐴 (,) 𝐵 ) )
880 835 840 846 879 syl21anc ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → 𝑦 ∈ ( 𝐴 (,) 𝐵 ) )
881 833 880 elind ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) )
882 881 3adantll2 ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) )
883 165 a1i ⊢ ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑡 ∈ ( 𝐺 “ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ) ) → Fun 𝐺 )
884 fvelima ⊢ ( ( Fun 𝐺 ∧ 𝑡 ∈ ( 𝐺 “ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ) ) → ∃ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ( 𝐺 ‘ 𝑦 ) = 𝑡 )
885 883 884 sylancom ⊢ ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑡 ∈ ( 𝐺 “ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ) ) → ∃ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ( 𝐺 ‘ 𝑦 ) = 𝑡 )
886 simp3 ⊢ ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ∧ ( 𝐺 ‘ 𝑦 ) = 𝑡 ) → ( 𝐺 ‘ 𝑦 ) = 𝑡 )
887 simp1l ⊢ ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ∧ ( 𝐺 ‘ 𝑦 ) = 𝑡 ) → 𝜑 )
888 inss2 ⊢ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ⊆ ( 𝐴 (,) 𝐵 )
889 ioossicc ⊢ ( 𝐴 (,) 𝐵 ) ⊆ ( 𝐴 [,] 𝐵 )
890 888 889 sstri ⊢ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ⊆ ( 𝐴 [,] 𝐵 )
891 890 sseli ⊢ ( 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) → 𝑦 ∈ ( 𝐴 [,] 𝐵 ) )
892 891 3ad2ant2 ⊢ ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ∧ ( 𝐺 ‘ 𝑦 ) = 𝑡 ) → 𝑦 ∈ ( 𝐴 [,] 𝐵 ) )
893 887 892 137 syl2anc ⊢ ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ∧ ( 𝐺 ‘ 𝑦 ) = 𝑡 ) → ( 𝐺 ‘ 𝑦 ) = ( 𝐹 ‘ 𝑦 ) )
894 sslin ⊢ ( ( 𝐴 (,) 𝐵 ) ⊆ ( 𝐴 [,] 𝐵 ) → ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ⊆ ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) )
895 889 894 ax-mp ⊢ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ⊆ ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) )
896 895 sseli ⊢ ( 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) → 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) )
897 896 adantl ⊢ ( ( ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ) → 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) )
898 195 adantr ⊢ ( ( ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ) → ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) = ( ◡ 𝐹 “ 𝑢 ) )
899 897 898 eleqtrd ⊢ ( ( ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ) → 𝑦 ∈ ( ◡ 𝐹 “ 𝑢 ) )
900 899 adantll ⊢ ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ) → 𝑦 ∈ ( ◡ 𝐹 “ 𝑢 ) )
901 20 ad2antrr ⊢ ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ) → 𝐹 Fn ( 𝐴 [,] 𝐵 ) )
902 901 143 syl ⊢ ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ) → ( 𝑦 ∈ ( ◡ 𝐹 “ 𝑢 ) ↔ ( 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ∧ ( 𝐹 ‘ 𝑦 ) ∈ 𝑢 ) ) )
903 900 902 mpbid ⊢ ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ) → ( 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ∧ ( 𝐹 ‘ 𝑦 ) ∈ 𝑢 ) )
904 903 simprd ⊢ ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ) → ( 𝐹 ‘ 𝑦 ) ∈ 𝑢 )
905 904 3adant3 ⊢ ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ∧ ( 𝐺 ‘ 𝑦 ) = 𝑡 ) → ( 𝐹 ‘ 𝑦 ) ∈ 𝑢 )
906 893 905 eqeltrd ⊢ ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ∧ ( 𝐺 ‘ 𝑦 ) = 𝑡 ) → ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 )
907 886 906 eqeltrrd ⊢ ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ∧ ( 𝐺 ‘ 𝑦 ) = 𝑡 ) → 𝑡 ∈ 𝑢 )
908 907 3exp ⊢ ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) → ( 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) → ( ( 𝐺 ‘ 𝑦 ) = 𝑡 → 𝑡 ∈ 𝑢 ) ) )
909 908 adantr ⊢ ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑡 ∈ ( 𝐺 “ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ) ) → ( 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) → ( ( 𝐺 ‘ 𝑦 ) = 𝑡 → 𝑡 ∈ 𝑢 ) ) )
910 909 rexlimdv ⊢ ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑡 ∈ ( 𝐺 “ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ) ) → ( ∃ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ( 𝐺 ‘ 𝑦 ) = 𝑡 → 𝑡 ∈ 𝑢 ) )
911 885 910 mpd ⊢ ( ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ 𝑡 ∈ ( 𝐺 “ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ) ) → 𝑡 ∈ 𝑢 )
912 911 ralrimiva ⊢ ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) → ∀ 𝑡 ∈ ( 𝐺 “ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ) 𝑡 ∈ 𝑢 )
913 dfss3 ⊢ ( ( 𝐺 “ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ) ⊆ 𝑢 ↔ ∀ 𝑡 ∈ ( 𝐺 “ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ) 𝑡 ∈ 𝑢 )
914 912 913 sylibr ⊢ ( ( 𝜑 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) → ( 𝐺 “ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ) ⊆ 𝑢 )
915 914 ad4ant14 ⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) → ( 𝐺 “ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ) ⊆ 𝑢 )
916 915 3adant2 ⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) → ( 𝐺 “ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ) ⊆ 𝑢 )
917 916 ad2antrr ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ( 𝐺 “ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ) ⊆ 𝑢 )
918 eleq2 ⊢ ( 𝑣 = ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) → ( 𝑦 ∈ 𝑣 ↔ 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ) )
919 imaeq2 ⊢ ( 𝑣 = ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) → ( 𝐺 “ 𝑣 ) = ( 𝐺 “ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ) )
920 919 sseq1d ⊢ ( 𝑣 = ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) → ( ( 𝐺 “ 𝑣 ) ⊆ 𝑢 ↔ ( 𝐺 “ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ) ⊆ 𝑢 ) )
921 918 920 anbi12d ⊢ ( 𝑣 = ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) → ( ( 𝑦 ∈ 𝑣 ∧ ( 𝐺 “ 𝑣 ) ⊆ 𝑢 ) ↔ ( 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ∧ ( 𝐺 “ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ) ⊆ 𝑢 ) ) )
922 921 rspcev ⊢ ( ( ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ∈ 𝐽 ∧ ( 𝑦 ∈ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ∧ ( 𝐺 “ ( 𝑤 ∩ ( 𝐴 (,) 𝐵 ) ) ) ⊆ 𝑢 ) ) → ∃ 𝑣 ∈ 𝐽 ( 𝑦 ∈ 𝑣 ∧ ( 𝐺 “ 𝑣 ) ⊆ 𝑢 ) )
923 812 882 917 922 syl12anc ⊢ ( ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) ∧ ¬ ( 𝐹 ‘ 𝐵 ) ∈ 𝑢 ) → ∃ 𝑣 ∈ 𝐽 ( 𝑦 ∈ 𝑣 ∧ ( 𝐺 “ 𝑣 ) ⊆ 𝑢 ) )
924 806 923 pm2.61dan ⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) ∧ ¬ ( 𝐹 ‘ 𝐴 ) ∈ 𝑢 ) → ∃ 𝑣 ∈ 𝐽 ( 𝑦 ∈ 𝑣 ∧ ( 𝐺 “ 𝑣 ) ⊆ 𝑢 ) )
925 596 924 pm2.61dan ⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) → ∃ 𝑣 ∈ 𝐽 ( 𝑦 ∈ 𝑣 ∧ ( 𝐺 “ 𝑣 ) ⊆ 𝑢 ) )
926 93 925 syld3an1 ⊢ ( ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑢 ∈ ( 𝐾 ↾t ran 𝐹 ) ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) ∧ 𝑤 ∈ 𝐽 ∧ ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) ) → ∃ 𝑣 ∈ 𝐽 ( 𝑦 ∈ 𝑣 ∧ ( 𝐺 “ 𝑣 ) ⊆ 𝑢 ) )
927 926 rexlimdv3a ⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑢 ∈ ( 𝐾 ↾t ran 𝐹 ) ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) → ( ∃ 𝑤 ∈ 𝐽 ( ◡ 𝐹 “ 𝑢 ) = ( 𝑤 ∩ ( 𝐴 [,] 𝐵 ) ) → ∃ 𝑣 ∈ 𝐽 ( 𝑦 ∈ 𝑣 ∧ ( 𝐺 “ 𝑣 ) ⊆ 𝑢 ) ) )
928 88 927 mpd ⊢ ( ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑢 ∈ ( 𝐾 ↾t ran 𝐹 ) ) ∧ ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 ) → ∃ 𝑣 ∈ 𝐽 ( 𝑦 ∈ 𝑣 ∧ ( 𝐺 “ 𝑣 ) ⊆ 𝑢 ) )
929 928 ex ⊢ ( ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) ∧ 𝑢 ∈ ( 𝐾 ↾t ran 𝐹 ) ) → ( ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 → ∃ 𝑣 ∈ 𝐽 ( 𝑦 ∈ 𝑣 ∧ ( 𝐺 “ 𝑣 ) ⊆ 𝑢 ) ) )
930 929 ralrimiva ⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → ∀ 𝑢 ∈ ( 𝐾 ↾t ran 𝐹 ) ( ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 → ∃ 𝑣 ∈ 𝐽 ( 𝑦 ∈ 𝑣 ∧ ( 𝐺 “ 𝑣 ) ⊆ 𝑢 ) ) )
931 11 a1i ⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → 𝐽 ∈ ( TopOn ‘ ℝ ) )
932 resttopon ⊢ ( ( 𝐾 ∈ ( TopOn ‘ 𝑌 ) ∧ ran 𝐹 ⊆ 𝑌 ) → ( 𝐾 ↾t ran 𝐹 ) ∈ ( TopOn ‘ ran 𝐹 ) )
933 17 71 932 syl2anc ⊢ ( 𝜑 → ( 𝐾 ↾t ran 𝐹 ) ∈ ( TopOn ‘ ran 𝐹 ) )
934 933 adantr ⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → ( 𝐾 ↾t ran 𝐹 ) ∈ ( TopOn ‘ ran 𝐹 ) )
935 iscnp ⊢ ( ( 𝐽 ∈ ( TopOn ‘ ℝ ) ∧ ( 𝐾 ↾t ran 𝐹 ) ∈ ( TopOn ‘ ran 𝐹 ) ∧ 𝑦 ∈ ℝ ) → ( 𝐺 ∈ ( ( 𝐽 CnP ( 𝐾 ↾t ran 𝐹 ) ) ‘ 𝑦 ) ↔ ( 𝐺 : ℝ ⟶ ran 𝐹 ∧ ∀ 𝑢 ∈ ( 𝐾 ↾t ran 𝐹 ) ( ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 → ∃ 𝑣 ∈ 𝐽 ( 𝑦 ∈ 𝑣 ∧ ( 𝐺 “ 𝑣 ) ⊆ 𝑢 ) ) ) ) )
936 931 934 466 935 syl3anc ⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → ( 𝐺 ∈ ( ( 𝐽 CnP ( 𝐾 ↾t ran 𝐹 ) ) ‘ 𝑦 ) ↔ ( 𝐺 : ℝ ⟶ ran 𝐹 ∧ ∀ 𝑢 ∈ ( 𝐾 ↾t ran 𝐹 ) ( ( 𝐺 ‘ 𝑦 ) ∈ 𝑢 → ∃ 𝑣 ∈ 𝐽 ( 𝑦 ∈ 𝑣 ∧ ( 𝐺 “ 𝑣 ) ⊆ 𝑢 ) ) ) ) )
937 66 930 936 mpbir2and ⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ) → 𝐺 ∈ ( ( 𝐽 CnP ( 𝐾 ↾t ran 𝐹 ) ) ‘ 𝑦 ) )
938 937 ralrimiva ⊢ ( 𝜑 → ∀ 𝑦 ∈ ℝ 𝐺 ∈ ( ( 𝐽 CnP ( 𝐾 ↾t ran 𝐹 ) ) ‘ 𝑦 ) )
939 cncnp ⊢ ( ( 𝐽 ∈ ( TopOn ‘ ℝ ) ∧ ( 𝐾 ↾t ran 𝐹 ) ∈ ( TopOn ‘ ran 𝐹 ) ) → ( 𝐺 ∈ ( 𝐽 Cn ( 𝐾 ↾t ran 𝐹 ) ) ↔ ( 𝐺 : ℝ ⟶ ran 𝐹 ∧ ∀ 𝑦 ∈ ℝ 𝐺 ∈ ( ( 𝐽 CnP ( 𝐾 ↾t ran 𝐹 ) ) ‘ 𝑦 ) ) ) )
940 11 933 939 sylancr ⊢ ( 𝜑 → ( 𝐺 ∈ ( 𝐽 Cn ( 𝐾 ↾t ran 𝐹 ) ) ↔ ( 𝐺 : ℝ ⟶ ran 𝐹 ∧ ∀ 𝑦 ∈ ℝ 𝐺 ∈ ( ( 𝐽 CnP ( 𝐾 ↾t ran 𝐹 ) ) ‘ 𝑦 ) ) ) )
941 65 938 940 mpbir2and ⊢ ( 𝜑 → 𝐺 ∈ ( 𝐽 Cn ( 𝐾 ↾t ran 𝐹 ) ) )
942 fnssres ⊢ ( ( 𝐺 Fn ℝ ∧ ( 𝐴 [,] 𝐵 ) ⊆ ℝ ) → ( 𝐺 ↾ ( 𝐴 [,] 𝐵 ) ) Fn ( 𝐴 [,] 𝐵 ) )
943 504 12 942 syl2anc ⊢ ( 𝜑 → ( 𝐺 ↾ ( 𝐴 [,] 𝐵 ) ) Fn ( 𝐴 [,] 𝐵 ) )
944 fvres ⊢ ( 𝑦 ∈ ( 𝐴 [,] 𝐵 ) → ( ( 𝐺 ↾ ( 𝐴 [,] 𝐵 ) ) ‘ 𝑦 ) = ( 𝐺 ‘ 𝑦 ) )
945 944 adantl ⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → ( ( 𝐺 ↾ ( 𝐴 [,] 𝐵 ) ) ‘ 𝑦 ) = ( 𝐺 ‘ 𝑦 ) )
946 945 137 eqtrd ⊢ ( ( 𝜑 ∧ 𝑦 ∈ ( 𝐴 [,] 𝐵 ) ) → ( ( 𝐺 ↾ ( 𝐴 [,] 𝐵 ) ) ‘ 𝑦 ) = ( 𝐹 ‘ 𝑦 ) )
947 943 20 946 eqfnfvd ⊢ ( 𝜑 → ( 𝐺 ↾ ( 𝐴 [,] 𝐵 ) ) = 𝐹 )
948 941 947 jca ⊢ ( 𝜑 → ( 𝐺 ∈ ( 𝐽 Cn ( 𝐾 ↾t ran 𝐹 ) ) ∧ ( 𝐺 ↾ ( 𝐴 [,] 𝐵 ) ) = 𝐹 ) )