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Metamath Proof Explorer

Theorem 3wlkdlem6

Description: Lemma 6 for 3wlkd . (Contributed by AV, 7-Feb-2021)

Ref Expression
Hypotheses 3wlkd.p ⊢ 𝑃 = ⟨“ 𝐴 𝐵 𝐶 𝐷 ”⟩
3wlkd.f ⊢ 𝐹 = ⟨“ 𝐽 𝐾 𝐿 ”⟩
3wlkd.s ⊢ ( 𝜑 → ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉 ) ) )
3wlkd.n ⊢ ( 𝜑 → ( ( 𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ) ∧ ( 𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷 ) ∧ 𝐶 ≠ 𝐷 ) )
3wlkd.e ⊢ ( 𝜑 → ( { 𝐴 , 𝐵 } ⊆ ( 𝐼 ‘ 𝐽 ) ∧ { 𝐵 , 𝐶 } ⊆ ( 𝐼 ‘ 𝐾 ) ∧ { 𝐶 , 𝐷 } ⊆ ( 𝐼 ‘ 𝐿 ) ) )
Assertion 3wlkdlem6 ( 𝜑 → ( 𝐴 ∈ ( 𝐼 ‘ 𝐽 ) ∧ 𝐵 ∈ ( 𝐼 ‘ 𝐾 ) ∧ 𝐶 ∈ ( 𝐼 ‘ 𝐿 ) ) )

Proof

Step Hyp Ref Expression
1 3wlkd.p ⊢ 𝑃 = ⟨“ 𝐴 𝐵 𝐶 𝐷 ”⟩
2 3wlkd.f ⊢ 𝐹 = ⟨“ 𝐽 𝐾 𝐿 ”⟩
3 3wlkd.s ⊢ ( 𝜑 → ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉 ) ) )
4 3wlkd.n ⊢ ( 𝜑 → ( ( 𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ) ∧ ( 𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷 ) ∧ 𝐶 ≠ 𝐷 ) )
5 3wlkd.e ⊢ ( 𝜑 → ( { 𝐴 , 𝐵 } ⊆ ( 𝐼 ‘ 𝐽 ) ∧ { 𝐵 , 𝐶 } ⊆ ( 𝐼 ‘ 𝐾 ) ∧ { 𝐶 , 𝐷 } ⊆ ( 𝐼 ‘ 𝐿 ) ) )
6 1 2 3 3wlkdlem3 ⊢ ( 𝜑 → ( ( ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ 1 ) = 𝐵 ) ∧ ( ( 𝑃 ‘ 2 ) = 𝐶 ∧ ( 𝑃 ‘ 3 ) = 𝐷 ) ) )
7 preq12 ⊢ ( ( ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ 1 ) = 𝐵 ) → { ( 𝑃 ‘ 0 ) , ( 𝑃 ‘ 1 ) } = { 𝐴 , 𝐵 } )
8 7 sseq1d ⊢ ( ( ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ 1 ) = 𝐵 ) → ( { ( 𝑃 ‘ 0 ) , ( 𝑃 ‘ 1 ) } ⊆ ( 𝐼 ‘ 𝐽 ) ↔ { 𝐴 , 𝐵 } ⊆ ( 𝐼 ‘ 𝐽 ) ) )
9 8 adantr ⊢ ( ( ( ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ 1 ) = 𝐵 ) ∧ ( ( 𝑃 ‘ 2 ) = 𝐶 ∧ ( 𝑃 ‘ 3 ) = 𝐷 ) ) → ( { ( 𝑃 ‘ 0 ) , ( 𝑃 ‘ 1 ) } ⊆ ( 𝐼 ‘ 𝐽 ) ↔ { 𝐴 , 𝐵 } ⊆ ( 𝐼 ‘ 𝐽 ) ) )
10 preq12 ⊢ ( ( ( 𝑃 ‘ 1 ) = 𝐵 ∧ ( 𝑃 ‘ 2 ) = 𝐶 ) → { ( 𝑃 ‘ 1 ) , ( 𝑃 ‘ 2 ) } = { 𝐵 , 𝐶 } )
11 10 ad2ant2lr ⊢ ( ( ( ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ 1 ) = 𝐵 ) ∧ ( ( 𝑃 ‘ 2 ) = 𝐶 ∧ ( 𝑃 ‘ 3 ) = 𝐷 ) ) → { ( 𝑃 ‘ 1 ) , ( 𝑃 ‘ 2 ) } = { 𝐵 , 𝐶 } )
12 11 sseq1d ⊢ ( ( ( ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ 1 ) = 𝐵 ) ∧ ( ( 𝑃 ‘ 2 ) = 𝐶 ∧ ( 𝑃 ‘ 3 ) = 𝐷 ) ) → ( { ( 𝑃 ‘ 1 ) , ( 𝑃 ‘ 2 ) } ⊆ ( 𝐼 ‘ 𝐾 ) ↔ { 𝐵 , 𝐶 } ⊆ ( 𝐼 ‘ 𝐾 ) ) )
13 preq12 ⊢ ( ( ( 𝑃 ‘ 2 ) = 𝐶 ∧ ( 𝑃 ‘ 3 ) = 𝐷 ) → { ( 𝑃 ‘ 2 ) , ( 𝑃 ‘ 3 ) } = { 𝐶 , 𝐷 } )
14 13 sseq1d ⊢ ( ( ( 𝑃 ‘ 2 ) = 𝐶 ∧ ( 𝑃 ‘ 3 ) = 𝐷 ) → ( { ( 𝑃 ‘ 2 ) , ( 𝑃 ‘ 3 ) } ⊆ ( 𝐼 ‘ 𝐿 ) ↔ { 𝐶 , 𝐷 } ⊆ ( 𝐼 ‘ 𝐿 ) ) )
15 14 adantl ⊢ ( ( ( ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ 1 ) = 𝐵 ) ∧ ( ( 𝑃 ‘ 2 ) = 𝐶 ∧ ( 𝑃 ‘ 3 ) = 𝐷 ) ) → ( { ( 𝑃 ‘ 2 ) , ( 𝑃 ‘ 3 ) } ⊆ ( 𝐼 ‘ 𝐿 ) ↔ { 𝐶 , 𝐷 } ⊆ ( 𝐼 ‘ 𝐿 ) ) )
16 9 12 15 3anbi123d ⊢ ( ( ( ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ 1 ) = 𝐵 ) ∧ ( ( 𝑃 ‘ 2 ) = 𝐶 ∧ ( 𝑃 ‘ 3 ) = 𝐷 ) ) → ( ( { ( 𝑃 ‘ 0 ) , ( 𝑃 ‘ 1 ) } ⊆ ( 𝐼 ‘ 𝐽 ) ∧ { ( 𝑃 ‘ 1 ) , ( 𝑃 ‘ 2 ) } ⊆ ( 𝐼 ‘ 𝐾 ) ∧ { ( 𝑃 ‘ 2 ) , ( 𝑃 ‘ 3 ) } ⊆ ( 𝐼 ‘ 𝐿 ) ) ↔ ( { 𝐴 , 𝐵 } ⊆ ( 𝐼 ‘ 𝐽 ) ∧ { 𝐵 , 𝐶 } ⊆ ( 𝐼 ‘ 𝐾 ) ∧ { 𝐶 , 𝐷 } ⊆ ( 𝐼 ‘ 𝐿 ) ) ) )
17 5 16 syl5ibrcom ⊢ ( 𝜑 → ( ( ( ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ 1 ) = 𝐵 ) ∧ ( ( 𝑃 ‘ 2 ) = 𝐶 ∧ ( 𝑃 ‘ 3 ) = 𝐷 ) ) → ( { ( 𝑃 ‘ 0 ) , ( 𝑃 ‘ 1 ) } ⊆ ( 𝐼 ‘ 𝐽 ) ∧ { ( 𝑃 ‘ 1 ) , ( 𝑃 ‘ 2 ) } ⊆ ( 𝐼 ‘ 𝐾 ) ∧ { ( 𝑃 ‘ 2 ) , ( 𝑃 ‘ 3 ) } ⊆ ( 𝐼 ‘ 𝐿 ) ) ) )
18 6 17 mpd ⊢ ( 𝜑 → ( { ( 𝑃 ‘ 0 ) , ( 𝑃 ‘ 1 ) } ⊆ ( 𝐼 ‘ 𝐽 ) ∧ { ( 𝑃 ‘ 1 ) , ( 𝑃 ‘ 2 ) } ⊆ ( 𝐼 ‘ 𝐾 ) ∧ { ( 𝑃 ‘ 2 ) , ( 𝑃 ‘ 3 ) } ⊆ ( 𝐼 ‘ 𝐿 ) ) )
19 fvex ⊢ ( 𝑃 ‘ 0 ) ∈ V
20 fvex ⊢ ( 𝑃 ‘ 1 ) ∈ V
21 19 20 prss ⊢ ( ( ( 𝑃 ‘ 0 ) ∈ ( 𝐼 ‘ 𝐽 ) ∧ ( 𝑃 ‘ 1 ) ∈ ( 𝐼 ‘ 𝐽 ) ) ↔ { ( 𝑃 ‘ 0 ) , ( 𝑃 ‘ 1 ) } ⊆ ( 𝐼 ‘ 𝐽 ) )
22 simpl ⊢ ( ( ( 𝑃 ‘ 0 ) ∈ ( 𝐼 ‘ 𝐽 ) ∧ ( 𝑃 ‘ 1 ) ∈ ( 𝐼 ‘ 𝐽 ) ) → ( 𝑃 ‘ 0 ) ∈ ( 𝐼 ‘ 𝐽 ) )
23 21 22 sylbir ⊢ ( { ( 𝑃 ‘ 0 ) , ( 𝑃 ‘ 1 ) } ⊆ ( 𝐼 ‘ 𝐽 ) → ( 𝑃 ‘ 0 ) ∈ ( 𝐼 ‘ 𝐽 ) )
24 fvex ⊢ ( 𝑃 ‘ 2 ) ∈ V
25 20 24 prss ⊢ ( ( ( 𝑃 ‘ 1 ) ∈ ( 𝐼 ‘ 𝐾 ) ∧ ( 𝑃 ‘ 2 ) ∈ ( 𝐼 ‘ 𝐾 ) ) ↔ { ( 𝑃 ‘ 1 ) , ( 𝑃 ‘ 2 ) } ⊆ ( 𝐼 ‘ 𝐾 ) )
26 simpl ⊢ ( ( ( 𝑃 ‘ 1 ) ∈ ( 𝐼 ‘ 𝐾 ) ∧ ( 𝑃 ‘ 2 ) ∈ ( 𝐼 ‘ 𝐾 ) ) → ( 𝑃 ‘ 1 ) ∈ ( 𝐼 ‘ 𝐾 ) )
27 25 26 sylbir ⊢ ( { ( 𝑃 ‘ 1 ) , ( 𝑃 ‘ 2 ) } ⊆ ( 𝐼 ‘ 𝐾 ) → ( 𝑃 ‘ 1 ) ∈ ( 𝐼 ‘ 𝐾 ) )
28 fvex ⊢ ( 𝑃 ‘ 3 ) ∈ V
29 24 28 prss ⊢ ( ( ( 𝑃 ‘ 2 ) ∈ ( 𝐼 ‘ 𝐿 ) ∧ ( 𝑃 ‘ 3 ) ∈ ( 𝐼 ‘ 𝐿 ) ) ↔ { ( 𝑃 ‘ 2 ) , ( 𝑃 ‘ 3 ) } ⊆ ( 𝐼 ‘ 𝐿 ) )
30 simpl ⊢ ( ( ( 𝑃 ‘ 2 ) ∈ ( 𝐼 ‘ 𝐿 ) ∧ ( 𝑃 ‘ 3 ) ∈ ( 𝐼 ‘ 𝐿 ) ) → ( 𝑃 ‘ 2 ) ∈ ( 𝐼 ‘ 𝐿 ) )
31 29 30 sylbir ⊢ ( { ( 𝑃 ‘ 2 ) , ( 𝑃 ‘ 3 ) } ⊆ ( 𝐼 ‘ 𝐿 ) → ( 𝑃 ‘ 2 ) ∈ ( 𝐼 ‘ 𝐿 ) )
32 23 27 31 3anim123i ⊢ ( ( { ( 𝑃 ‘ 0 ) , ( 𝑃 ‘ 1 ) } ⊆ ( 𝐼 ‘ 𝐽 ) ∧ { ( 𝑃 ‘ 1 ) , ( 𝑃 ‘ 2 ) } ⊆ ( 𝐼 ‘ 𝐾 ) ∧ { ( 𝑃 ‘ 2 ) , ( 𝑃 ‘ 3 ) } ⊆ ( 𝐼 ‘ 𝐿 ) ) → ( ( 𝑃 ‘ 0 ) ∈ ( 𝐼 ‘ 𝐽 ) ∧ ( 𝑃 ‘ 1 ) ∈ ( 𝐼 ‘ 𝐾 ) ∧ ( 𝑃 ‘ 2 ) ∈ ( 𝐼 ‘ 𝐿 ) ) )
33 18 32 syl ⊢ ( 𝜑 → ( ( 𝑃 ‘ 0 ) ∈ ( 𝐼 ‘ 𝐽 ) ∧ ( 𝑃 ‘ 1 ) ∈ ( 𝐼 ‘ 𝐾 ) ∧ ( 𝑃 ‘ 2 ) ∈ ( 𝐼 ‘ 𝐿 ) ) )
34 eleq1 ⊢ ( ( 𝑃 ‘ 0 ) = 𝐴 → ( ( 𝑃 ‘ 0 ) ∈ ( 𝐼 ‘ 𝐽 ) ↔ 𝐴 ∈ ( 𝐼 ‘ 𝐽 ) ) )
35 34 adantr ⊢ ( ( ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ 1 ) = 𝐵 ) → ( ( 𝑃 ‘ 0 ) ∈ ( 𝐼 ‘ 𝐽 ) ↔ 𝐴 ∈ ( 𝐼 ‘ 𝐽 ) ) )
36 35 adantr ⊢ ( ( ( ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ 1 ) = 𝐵 ) ∧ ( ( 𝑃 ‘ 2 ) = 𝐶 ∧ ( 𝑃 ‘ 3 ) = 𝐷 ) ) → ( ( 𝑃 ‘ 0 ) ∈ ( 𝐼 ‘ 𝐽 ) ↔ 𝐴 ∈ ( 𝐼 ‘ 𝐽 ) ) )
37 eleq1 ⊢ ( ( 𝑃 ‘ 1 ) = 𝐵 → ( ( 𝑃 ‘ 1 ) ∈ ( 𝐼 ‘ 𝐾 ) ↔ 𝐵 ∈ ( 𝐼 ‘ 𝐾 ) ) )
38 37 adantl ⊢ ( ( ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ 1 ) = 𝐵 ) → ( ( 𝑃 ‘ 1 ) ∈ ( 𝐼 ‘ 𝐾 ) ↔ 𝐵 ∈ ( 𝐼 ‘ 𝐾 ) ) )
39 38 adantr ⊢ ( ( ( ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ 1 ) = 𝐵 ) ∧ ( ( 𝑃 ‘ 2 ) = 𝐶 ∧ ( 𝑃 ‘ 3 ) = 𝐷 ) ) → ( ( 𝑃 ‘ 1 ) ∈ ( 𝐼 ‘ 𝐾 ) ↔ 𝐵 ∈ ( 𝐼 ‘ 𝐾 ) ) )
40 eleq1 ⊢ ( ( 𝑃 ‘ 2 ) = 𝐶 → ( ( 𝑃 ‘ 2 ) ∈ ( 𝐼 ‘ 𝐿 ) ↔ 𝐶 ∈ ( 𝐼 ‘ 𝐿 ) ) )
41 40 adantr ⊢ ( ( ( 𝑃 ‘ 2 ) = 𝐶 ∧ ( 𝑃 ‘ 3 ) = 𝐷 ) → ( ( 𝑃 ‘ 2 ) ∈ ( 𝐼 ‘ 𝐿 ) ↔ 𝐶 ∈ ( 𝐼 ‘ 𝐿 ) ) )
42 41 adantl ⊢ ( ( ( ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ 1 ) = 𝐵 ) ∧ ( ( 𝑃 ‘ 2 ) = 𝐶 ∧ ( 𝑃 ‘ 3 ) = 𝐷 ) ) → ( ( 𝑃 ‘ 2 ) ∈ ( 𝐼 ‘ 𝐿 ) ↔ 𝐶 ∈ ( 𝐼 ‘ 𝐿 ) ) )
43 36 39 42 3anbi123d ⊢ ( ( ( ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ 1 ) = 𝐵 ) ∧ ( ( 𝑃 ‘ 2 ) = 𝐶 ∧ ( 𝑃 ‘ 3 ) = 𝐷 ) ) → ( ( ( 𝑃 ‘ 0 ) ∈ ( 𝐼 ‘ 𝐽 ) ∧ ( 𝑃 ‘ 1 ) ∈ ( 𝐼 ‘ 𝐾 ) ∧ ( 𝑃 ‘ 2 ) ∈ ( 𝐼 ‘ 𝐿 ) ) ↔ ( 𝐴 ∈ ( 𝐼 ‘ 𝐽 ) ∧ 𝐵 ∈ ( 𝐼 ‘ 𝐾 ) ∧ 𝐶 ∈ ( 𝐼 ‘ 𝐿 ) ) ) )
44 43 bicomd ⊢ ( ( ( ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ 1 ) = 𝐵 ) ∧ ( ( 𝑃 ‘ 2 ) = 𝐶 ∧ ( 𝑃 ‘ 3 ) = 𝐷 ) ) → ( ( 𝐴 ∈ ( 𝐼 ‘ 𝐽 ) ∧ 𝐵 ∈ ( 𝐼 ‘ 𝐾 ) ∧ 𝐶 ∈ ( 𝐼 ‘ 𝐿 ) ) ↔ ( ( 𝑃 ‘ 0 ) ∈ ( 𝐼 ‘ 𝐽 ) ∧ ( 𝑃 ‘ 1 ) ∈ ( 𝐼 ‘ 𝐾 ) ∧ ( 𝑃 ‘ 2 ) ∈ ( 𝐼 ‘ 𝐿 ) ) ) )
45 6 44 syl ⊢ ( 𝜑 → ( ( 𝐴 ∈ ( 𝐼 ‘ 𝐽 ) ∧ 𝐵 ∈ ( 𝐼 ‘ 𝐾 ) ∧ 𝐶 ∈ ( 𝐼 ‘ 𝐿 ) ) ↔ ( ( 𝑃 ‘ 0 ) ∈ ( 𝐼 ‘ 𝐽 ) ∧ ( 𝑃 ‘ 1 ) ∈ ( 𝐼 ‘ 𝐾 ) ∧ ( 𝑃 ‘ 2 ) ∈ ( 𝐼 ‘ 𝐿 ) ) ) )
46 33 45 mpbird ⊢ ( 𝜑 → ( 𝐴 ∈ ( 𝐼 ‘ 𝐽 ) ∧ 𝐵 ∈ ( 𝐼 ‘ 𝐾 ) ∧ 𝐶 ∈ ( 𝐼 ‘ 𝐿 ) ) )