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 ) ) )