Theorem 3wlkdlem9

Description: Lemma 9 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	3wlkdlem9	⊢ ( 𝜑 → ( { 𝐴 , 𝐵 } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 0 ) ) ∧ { 𝐵 , 𝐶 } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 1 ) ) ∧ { 𝐶 , 𝐷 } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 2 ) ) ) )

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 4 5	3wlkdlem8	⊢ ( 𝜑 → ( ( 𝐹 ‘ 0 ) = 𝐽 ∧ ( 𝐹 ‘ 1 ) = 𝐾 ∧ ( 𝐹 ‘ 2 ) = 𝐿 ) )
7		fveq2	⊢ ( ( 𝐹 ‘ 0 ) = 𝐽 → ( 𝐼 ‘ ( 𝐹 ‘ 0 ) ) = ( 𝐼 ‘ 𝐽 ) )
8	7	sseq2d	⊢ ( ( 𝐹 ‘ 0 ) = 𝐽 → ( { 𝐴 , 𝐵 } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 0 ) ) ↔ { 𝐴 , 𝐵 } ⊆ ( 𝐼 ‘ 𝐽 ) ) )
9	8	3ad2ant1	⊢ ( ( ( 𝐹 ‘ 0 ) = 𝐽 ∧ ( 𝐹 ‘ 1 ) = 𝐾 ∧ ( 𝐹 ‘ 2 ) = 𝐿 ) → ( { 𝐴 , 𝐵 } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 0 ) ) ↔ { 𝐴 , 𝐵 } ⊆ ( 𝐼 ‘ 𝐽 ) ) )
10		fveq2	⊢ ( ( 𝐹 ‘ 1 ) = 𝐾 → ( 𝐼 ‘ ( 𝐹 ‘ 1 ) ) = ( 𝐼 ‘ 𝐾 ) )
11	10	sseq2d	⊢ ( ( 𝐹 ‘ 1 ) = 𝐾 → ( { 𝐵 , 𝐶 } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 1 ) ) ↔ { 𝐵 , 𝐶 } ⊆ ( 𝐼 ‘ 𝐾 ) ) )
12	11	3ad2ant2	⊢ ( ( ( 𝐹 ‘ 0 ) = 𝐽 ∧ ( 𝐹 ‘ 1 ) = 𝐾 ∧ ( 𝐹 ‘ 2 ) = 𝐿 ) → ( { 𝐵 , 𝐶 } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 1 ) ) ↔ { 𝐵 , 𝐶 } ⊆ ( 𝐼 ‘ 𝐾 ) ) )
13		fveq2	⊢ ( ( 𝐹 ‘ 2 ) = 𝐿 → ( 𝐼 ‘ ( 𝐹 ‘ 2 ) ) = ( 𝐼 ‘ 𝐿 ) )
14	13	sseq2d	⊢ ( ( 𝐹 ‘ 2 ) = 𝐿 → ( { 𝐶 , 𝐷 } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 2 ) ) ↔ { 𝐶 , 𝐷 } ⊆ ( 𝐼 ‘ 𝐿 ) ) )
15	14	3ad2ant3	⊢ ( ( ( 𝐹 ‘ 0 ) = 𝐽 ∧ ( 𝐹 ‘ 1 ) = 𝐾 ∧ ( 𝐹 ‘ 2 ) = 𝐿 ) → ( { 𝐶 , 𝐷 } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 2 ) ) ↔ { 𝐶 , 𝐷 } ⊆ ( 𝐼 ‘ 𝐿 ) ) )
16	9 12 15	3anbi123d	⊢ ( ( ( 𝐹 ‘ 0 ) = 𝐽 ∧ ( 𝐹 ‘ 1 ) = 𝐾 ∧ ( 𝐹 ‘ 2 ) = 𝐿 ) → ( ( { 𝐴 , 𝐵 } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 0 ) ) ∧ { 𝐵 , 𝐶 } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 1 ) ) ∧ { 𝐶 , 𝐷 } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 2 ) ) ) ↔ ( { 𝐴 , 𝐵 } ⊆ ( 𝐼 ‘ 𝐽 ) ∧ { 𝐵 , 𝐶 } ⊆ ( 𝐼 ‘ 𝐾 ) ∧ { 𝐶 , 𝐷 } ⊆ ( 𝐼 ‘ 𝐿 ) ) ) )
17	6 16	syl	⊢ ( 𝜑 → ( ( { 𝐴 , 𝐵 } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 0 ) ) ∧ { 𝐵 , 𝐶 } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 1 ) ) ∧ { 𝐶 , 𝐷 } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 2 ) ) ) ↔ ( { 𝐴 , 𝐵 } ⊆ ( 𝐼 ‘ 𝐽 ) ∧ { 𝐵 , 𝐶 } ⊆ ( 𝐼 ‘ 𝐾 ) ∧ { 𝐶 , 𝐷 } ⊆ ( 𝐼 ‘ 𝐿 ) ) ) )
18	5 17	mpbird	⊢ ( 𝜑 → ( { 𝐴 , 𝐵 } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 0 ) ) ∧ { 𝐵 , 𝐶 } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 1 ) ) ∧ { 𝐶 , 𝐷 } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 2 ) ) ) )