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Metamath Proof Explorer

Theorem upbdrech

Description: Choice of an upper bound for a nonempty bunded set (image set version). (Contributed by Glauco Siliprandi, 11-Dec-2019)

Ref Expression
Hypotheses upbdrech.a ⊢ ( 𝜑 → 𝐴 ≠ ∅ )
upbdrech.b ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ∈ ℝ )
upbdrech.bd ⊢ ( 𝜑 → ∃ 𝑦 ∈ ℝ ∀ 𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 )
upbdrech.c ⊢ 𝐶 = sup ( { 𝑧 ∣ ∃ 𝑥 ∈ 𝐴 𝑧 = 𝐵 } , ℝ , < )
Assertion upbdrech ( 𝜑 → ( 𝐶 ∈ ℝ ∧ ∀ 𝑥 ∈ 𝐴 𝐵 ≤ 𝐶 ) )

Proof

Step Hyp Ref Expression
1 upbdrech.a ⊢ ( 𝜑 → 𝐴 ≠ ∅ )
2 upbdrech.b ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ∈ ℝ )
3 upbdrech.bd ⊢ ( 𝜑 → ∃ 𝑦 ∈ ℝ ∀ 𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 )
4 upbdrech.c ⊢ 𝐶 = sup ( { 𝑧 ∣ ∃ 𝑥 ∈ 𝐴 𝑧 = 𝐵 } , ℝ , < )
5 2 ralrimiva ⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐴 𝐵 ∈ ℝ )
6 nfra1 ⊢ Ⅎ 𝑥 ∀ 𝑥 ∈ 𝐴 𝐵 ∈ ℝ
7 nfv ⊢ Ⅎ 𝑥 𝑧 ∈ ℝ
8 simp3 ⊢ ( ( ∀ 𝑥 ∈ 𝐴 𝐵 ∈ ℝ ∧ 𝑥 ∈ 𝐴 ∧ 𝑧 = 𝐵 ) → 𝑧 = 𝐵 )
9 rspa ⊢ ( ( ∀ 𝑥 ∈ 𝐴 𝐵 ∈ ℝ ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ∈ ℝ )
10 9 3adant3 ⊢ ( ( ∀ 𝑥 ∈ 𝐴 𝐵 ∈ ℝ ∧ 𝑥 ∈ 𝐴 ∧ 𝑧 = 𝐵 ) → 𝐵 ∈ ℝ )
11 8 10 eqeltrd ⊢ ( ( ∀ 𝑥 ∈ 𝐴 𝐵 ∈ ℝ ∧ 𝑥 ∈ 𝐴 ∧ 𝑧 = 𝐵 ) → 𝑧 ∈ ℝ )
12 11 3exp ⊢ ( ∀ 𝑥 ∈ 𝐴 𝐵 ∈ ℝ → ( 𝑥 ∈ 𝐴 → ( 𝑧 = 𝐵 → 𝑧 ∈ ℝ ) ) )
13 6 7 12 rexlimd ⊢ ( ∀ 𝑥 ∈ 𝐴 𝐵 ∈ ℝ → ( ∃ 𝑥 ∈ 𝐴 𝑧 = 𝐵 → 𝑧 ∈ ℝ ) )
14 13 abssdv ⊢ ( ∀ 𝑥 ∈ 𝐴 𝐵 ∈ ℝ → { 𝑧 ∣ ∃ 𝑥 ∈ 𝐴 𝑧 = 𝐵 } ⊆ ℝ )
15 5 14 syl ⊢ ( 𝜑 → { 𝑧 ∣ ∃ 𝑥 ∈ 𝐴 𝑧 = 𝐵 } ⊆ ℝ )
16 eqidd ⊢ ( 𝑥 ∈ 𝐴 → 𝐵 = 𝐵 )
17 16 rgen ⊢ ∀ 𝑥 ∈ 𝐴 𝐵 = 𝐵
18 r19.2z ⊢ ( ( 𝐴 ≠ ∅ ∧ ∀ 𝑥 ∈ 𝐴 𝐵 = 𝐵 ) → ∃ 𝑥 ∈ 𝐴 𝐵 = 𝐵 )
19 1 17 18 sylancl ⊢ ( 𝜑 → ∃ 𝑥 ∈ 𝐴 𝐵 = 𝐵 )
20 nfv ⊢ Ⅎ 𝑥 𝜑
21 nfre1 ⊢ Ⅎ 𝑥 ∃ 𝑥 ∈ 𝐴 𝑧 = 𝐵
22 21 nfex ⊢ Ⅎ 𝑥 ∃ 𝑧 ∃ 𝑥 ∈ 𝐴 𝑧 = 𝐵
23 simpr ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝑥 ∈ 𝐴 )
24 elex ⊢ ( 𝐵 ∈ ℝ → 𝐵 ∈ V )
25 2 24 syl ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ∈ V )
26 isset ⊢ ( 𝐵 ∈ V ↔ ∃ 𝑧 𝑧 = 𝐵 )
27 25 26 sylib ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ∃ 𝑧 𝑧 = 𝐵 )
28 rspe ⊢ ( ( 𝑥 ∈ 𝐴 ∧ ∃ 𝑧 𝑧 = 𝐵 ) → ∃ 𝑥 ∈ 𝐴 ∃ 𝑧 𝑧 = 𝐵 )
29 23 27 28 syl2anc ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ∃ 𝑥 ∈ 𝐴 ∃ 𝑧 𝑧 = 𝐵 )
30 rexcom4 ⊢ ( ∃ 𝑥 ∈ 𝐴 ∃ 𝑧 𝑧 = 𝐵 ↔ ∃ 𝑧 ∃ 𝑥 ∈ 𝐴 𝑧 = 𝐵 )
31 29 30 sylib ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ∃ 𝑧 ∃ 𝑥 ∈ 𝐴 𝑧 = 𝐵 )
32 31 3adant3 ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ∧ 𝐵 = 𝐵 ) → ∃ 𝑧 ∃ 𝑥 ∈ 𝐴 𝑧 = 𝐵 )
33 32 3exp ⊢ ( 𝜑 → ( 𝑥 ∈ 𝐴 → ( 𝐵 = 𝐵 → ∃ 𝑧 ∃ 𝑥 ∈ 𝐴 𝑧 = 𝐵 ) ) )
34 20 22 33 rexlimd ⊢ ( 𝜑 → ( ∃ 𝑥 ∈ 𝐴 𝐵 = 𝐵 → ∃ 𝑧 ∃ 𝑥 ∈ 𝐴 𝑧 = 𝐵 ) )
35 19 34 mpd ⊢ ( 𝜑 → ∃ 𝑧 ∃ 𝑥 ∈ 𝐴 𝑧 = 𝐵 )
36 abn0 ⊢ ( { 𝑧 ∣ ∃ 𝑥 ∈ 𝐴 𝑧 = 𝐵 } ≠ ∅ ↔ ∃ 𝑧 ∃ 𝑥 ∈ 𝐴 𝑧 = 𝐵 )
37 35 36 sylibr ⊢ ( 𝜑 → { 𝑧 ∣ ∃ 𝑥 ∈ 𝐴 𝑧 = 𝐵 } ≠ ∅ )
38 vex ⊢ 𝑤 ∈ V
39 eqeq1 ⊢ ( 𝑧 = 𝑤 → ( 𝑧 = 𝐵 ↔ 𝑤 = 𝐵 ) )
40 39 rexbidv ⊢ ( 𝑧 = 𝑤 → ( ∃ 𝑥 ∈ 𝐴 𝑧 = 𝐵 ↔ ∃ 𝑥 ∈ 𝐴 𝑤 = 𝐵 ) )
41 38 40 elab ⊢ ( 𝑤 ∈ { 𝑧 ∣ ∃ 𝑥 ∈ 𝐴 𝑧 = 𝐵 } ↔ ∃ 𝑥 ∈ 𝐴 𝑤 = 𝐵 )
42 41 biimpi ⊢ ( 𝑤 ∈ { 𝑧 ∣ ∃ 𝑥 ∈ 𝐴 𝑧 = 𝐵 } → ∃ 𝑥 ∈ 𝐴 𝑤 = 𝐵 )
43 42 adantl ⊢ ( ( ( 𝜑 ∧ ∀ 𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 ) ∧ 𝑤 ∈ { 𝑧 ∣ ∃ 𝑥 ∈ 𝐴 𝑧 = 𝐵 } ) → ∃ 𝑥 ∈ 𝐴 𝑤 = 𝐵 )
44 nfra1 ⊢ Ⅎ 𝑥 ∀ 𝑥 ∈ 𝐴 𝐵 ≤ 𝑦
45 20 44 nfan ⊢ Ⅎ 𝑥 ( 𝜑 ∧ ∀ 𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 )
46 21 nfsab ⊢ Ⅎ 𝑥 𝑤 ∈ { 𝑧 ∣ ∃ 𝑥 ∈ 𝐴 𝑧 = 𝐵 }
47 45 46 nfan ⊢ Ⅎ 𝑥 ( ( 𝜑 ∧ ∀ 𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 ) ∧ 𝑤 ∈ { 𝑧 ∣ ∃ 𝑥 ∈ 𝐴 𝑧 = 𝐵 } )
48 nfv ⊢ Ⅎ 𝑥 𝑤 ≤ 𝑦
49 simp3 ⊢ ( ( ( 𝜑 ∧ ∀ 𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 ) ∧ 𝑥 ∈ 𝐴 ∧ 𝑤 = 𝐵 ) → 𝑤 = 𝐵 )
50 simp1r ⊢ ( ( ( 𝜑 ∧ ∀ 𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 ) ∧ 𝑥 ∈ 𝐴 ∧ 𝑤 = 𝐵 ) → ∀ 𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 )
51 simp2 ⊢ ( ( ( 𝜑 ∧ ∀ 𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 ) ∧ 𝑥 ∈ 𝐴 ∧ 𝑤 = 𝐵 ) → 𝑥 ∈ 𝐴 )
52 rspa ⊢ ( ( ∀ 𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ≤ 𝑦 )
53 50 51 52 syl2anc ⊢ ( ( ( 𝜑 ∧ ∀ 𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 ) ∧ 𝑥 ∈ 𝐴 ∧ 𝑤 = 𝐵 ) → 𝐵 ≤ 𝑦 )
54 49 53 eqbrtrd ⊢ ( ( ( 𝜑 ∧ ∀ 𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 ) ∧ 𝑥 ∈ 𝐴 ∧ 𝑤 = 𝐵 ) → 𝑤 ≤ 𝑦 )
55 54 3exp ⊢ ( ( 𝜑 ∧ ∀ 𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 ) → ( 𝑥 ∈ 𝐴 → ( 𝑤 = 𝐵 → 𝑤 ≤ 𝑦 ) ) )
56 55 adantr ⊢ ( ( ( 𝜑 ∧ ∀ 𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 ) ∧ 𝑤 ∈ { 𝑧 ∣ ∃ 𝑥 ∈ 𝐴 𝑧 = 𝐵 } ) → ( 𝑥 ∈ 𝐴 → ( 𝑤 = 𝐵 → 𝑤 ≤ 𝑦 ) ) )
57 47 48 56 rexlimd ⊢ ( ( ( 𝜑 ∧ ∀ 𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 ) ∧ 𝑤 ∈ { 𝑧 ∣ ∃ 𝑥 ∈ 𝐴 𝑧 = 𝐵 } ) → ( ∃ 𝑥 ∈ 𝐴 𝑤 = 𝐵 → 𝑤 ≤ 𝑦 ) )
58 43 57 mpd ⊢ ( ( ( 𝜑 ∧ ∀ 𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 ) ∧ 𝑤 ∈ { 𝑧 ∣ ∃ 𝑥 ∈ 𝐴 𝑧 = 𝐵 } ) → 𝑤 ≤ 𝑦 )
59 58 ralrimiva ⊢ ( ( 𝜑 ∧ ∀ 𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 ) → ∀ 𝑤 ∈ { 𝑧 ∣ ∃ 𝑥 ∈ 𝐴 𝑧 = 𝐵 } 𝑤 ≤ 𝑦 )
60 59 3adant2 ⊢ ( ( 𝜑 ∧ 𝑦 ∈ ℝ ∧ ∀ 𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 ) → ∀ 𝑤 ∈ { 𝑧 ∣ ∃ 𝑥 ∈ 𝐴 𝑧 = 𝐵 } 𝑤 ≤ 𝑦 )
61 60 3exp ⊢ ( 𝜑 → ( 𝑦 ∈ ℝ → ( ∀ 𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 → ∀ 𝑤 ∈ { 𝑧 ∣ ∃ 𝑥 ∈ 𝐴 𝑧 = 𝐵 } 𝑤 ≤ 𝑦 ) ) )
62 61 reximdvai ⊢ ( 𝜑 → ( ∃ 𝑦 ∈ ℝ ∀ 𝑥 ∈ 𝐴 𝐵 ≤ 𝑦 → ∃ 𝑦 ∈ ℝ ∀ 𝑤 ∈ { 𝑧 ∣ ∃ 𝑥 ∈ 𝐴 𝑧 = 𝐵 } 𝑤 ≤ 𝑦 ) )
63 3 62 mpd ⊢ ( 𝜑 → ∃ 𝑦 ∈ ℝ ∀ 𝑤 ∈ { 𝑧 ∣ ∃ 𝑥 ∈ 𝐴 𝑧 = 𝐵 } 𝑤 ≤ 𝑦 )
64 suprcl ⊢ ( ( { 𝑧 ∣ ∃ 𝑥 ∈ 𝐴 𝑧 = 𝐵 } ⊆ ℝ ∧ { 𝑧 ∣ ∃ 𝑥 ∈ 𝐴 𝑧 = 𝐵 } ≠ ∅ ∧ ∃ 𝑦 ∈ ℝ ∀ 𝑤 ∈ { 𝑧 ∣ ∃ 𝑥 ∈ 𝐴 𝑧 = 𝐵 } 𝑤 ≤ 𝑦 ) → sup ( { 𝑧 ∣ ∃ 𝑥 ∈ 𝐴 𝑧 = 𝐵 } , ℝ , < ) ∈ ℝ )
65 15 37 63 64 syl3anc ⊢ ( 𝜑 → sup ( { 𝑧 ∣ ∃ 𝑥 ∈ 𝐴 𝑧 = 𝐵 } , ℝ , < ) ∈ ℝ )
66 4 65 eqeltrid ⊢ ( 𝜑 → 𝐶 ∈ ℝ )
67 15 adantr ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → { 𝑧 ∣ ∃ 𝑥 ∈ 𝐴 𝑧 = 𝐵 } ⊆ ℝ )
68 31 36 sylibr ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → { 𝑧 ∣ ∃ 𝑥 ∈ 𝐴 𝑧 = 𝐵 } ≠ ∅ )
69 63 adantr ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → ∃ 𝑦 ∈ ℝ ∀ 𝑤 ∈ { 𝑧 ∣ ∃ 𝑥 ∈ 𝐴 𝑧 = 𝐵 } 𝑤 ≤ 𝑦 )
70 elabrexg ⊢ ( ( 𝑥 ∈ 𝐴 ∧ 𝐵 ∈ ℝ ) → 𝐵 ∈ { 𝑧 ∣ ∃ 𝑥 ∈ 𝐴 𝑧 = 𝐵 } )
71 23 2 70 syl2anc ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ∈ { 𝑧 ∣ ∃ 𝑥 ∈ 𝐴 𝑧 = 𝐵 } )
72 suprub ⊢ ( ( ( { 𝑧 ∣ ∃ 𝑥 ∈ 𝐴 𝑧 = 𝐵 } ⊆ ℝ ∧ { 𝑧 ∣ ∃ 𝑥 ∈ 𝐴 𝑧 = 𝐵 } ≠ ∅ ∧ ∃ 𝑦 ∈ ℝ ∀ 𝑤 ∈ { 𝑧 ∣ ∃ 𝑥 ∈ 𝐴 𝑧 = 𝐵 } 𝑤 ≤ 𝑦 ) ∧ 𝐵 ∈ { 𝑧 ∣ ∃ 𝑥 ∈ 𝐴 𝑧 = 𝐵 } ) → 𝐵 ≤ sup ( { 𝑧 ∣ ∃ 𝑥 ∈ 𝐴 𝑧 = 𝐵 } , ℝ , < ) )
73 67 68 69 71 72 syl31anc ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ≤ sup ( { 𝑧 ∣ ∃ 𝑥 ∈ 𝐴 𝑧 = 𝐵 } , ℝ , < ) )
74 73 4 breqtrrdi ⊢ ( ( 𝜑 ∧ 𝑥 ∈ 𝐴 ) → 𝐵 ≤ 𝐶 )
75 74 ralrimiva ⊢ ( 𝜑 → ∀ 𝑥 ∈ 𝐴 𝐵 ≤ 𝐶 )
76 66 75 jca ⊢ ( 𝜑 → ( 𝐶 ∈ ℝ ∧ ∀ 𝑥 ∈ 𝐴 𝐵 ≤ 𝐶 ) )