Theorem 2wlkdlem9

Description: Lemma 9 for 2wlkd . (Contributed by AV, 14-Feb-2021)

	Ref	Expression
Hypotheses	2wlkd.p	⊢ 𝑃 = ⟨“ 𝐴 𝐵 𝐶 ”⟩
	2wlkd.f	⊢ 𝐹 = ⟨“ 𝐽 𝐾 ”⟩
	2wlkd.s	⊢ ( 𝜑 → ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ) )
	2wlkd.n	⊢ ( 𝜑 → ( 𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ) )
	2wlkd.e	⊢ ( 𝜑 → ( { 𝐴 , 𝐵 } ⊆ ( 𝐼 ‘ 𝐽 ) ∧ { 𝐵 , 𝐶 } ⊆ ( 𝐼 ‘ 𝐾 ) ) )
Assertion	2wlkdlem9	⊢ ( 𝜑 → ( { 𝐴 , 𝐵 } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 0 ) ) ∧ { 𝐵 , 𝐶 } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 1 ) ) ) )

Proof

Step	Hyp	Ref	Expression
1		2wlkd.p	⊢ 𝑃 = ⟨“ 𝐴 𝐵 𝐶 ”⟩
2		2wlkd.f	⊢ 𝐹 = ⟨“ 𝐽 𝐾 ”⟩
3		2wlkd.s	⊢ ( 𝜑 → ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ) )
4		2wlkd.n	⊢ ( 𝜑 → ( 𝐴 ≠ 𝐵 ∧ 𝐵 ≠ 𝐶 ) )
5		2wlkd.e	⊢ ( 𝜑 → ( { 𝐴 , 𝐵 } ⊆ ( 𝐼 ‘ 𝐽 ) ∧ { 𝐵 , 𝐶 } ⊆ ( 𝐼 ‘ 𝐾 ) ) )
6	1 2 3 4 5	2wlkdlem8	⊢ ( 𝜑 → ( ( 𝐹 ‘ 0 ) = 𝐽 ∧ ( 𝐹 ‘ 1 ) = 𝐾 ) )
7		fveq2	⊢ ( ( 𝐹 ‘ 0 ) = 𝐽 → ( 𝐼 ‘ ( 𝐹 ‘ 0 ) ) = ( 𝐼 ‘ 𝐽 ) )
8	7	adantr	⊢ ( ( ( 𝐹 ‘ 0 ) = 𝐽 ∧ ( 𝐹 ‘ 1 ) = 𝐾 ) → ( 𝐼 ‘ ( 𝐹 ‘ 0 ) ) = ( 𝐼 ‘ 𝐽 ) )
9	8	sseq2d	⊢ ( ( ( 𝐹 ‘ 0 ) = 𝐽 ∧ ( 𝐹 ‘ 1 ) = 𝐾 ) → ( { 𝐴 , 𝐵 } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 0 ) ) ↔ { 𝐴 , 𝐵 } ⊆ ( 𝐼 ‘ 𝐽 ) ) )
10		fveq2	⊢ ( ( 𝐹 ‘ 1 ) = 𝐾 → ( 𝐼 ‘ ( 𝐹 ‘ 1 ) ) = ( 𝐼 ‘ 𝐾 ) )
11	10	adantl	⊢ ( ( ( 𝐹 ‘ 0 ) = 𝐽 ∧ ( 𝐹 ‘ 1 ) = 𝐾 ) → ( 𝐼 ‘ ( 𝐹 ‘ 1 ) ) = ( 𝐼 ‘ 𝐾 ) )
12	11	sseq2d	⊢ ( ( ( 𝐹 ‘ 0 ) = 𝐽 ∧ ( 𝐹 ‘ 1 ) = 𝐾 ) → ( { 𝐵 , 𝐶 } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 1 ) ) ↔ { 𝐵 , 𝐶 } ⊆ ( 𝐼 ‘ 𝐾 ) ) )
13	9 12	anbi12d	⊢ ( ( ( 𝐹 ‘ 0 ) = 𝐽 ∧ ( 𝐹 ‘ 1 ) = 𝐾 ) → ( ( { 𝐴 , 𝐵 } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 0 ) ) ∧ { 𝐵 , 𝐶 } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 1 ) ) ) ↔ ( { 𝐴 , 𝐵 } ⊆ ( 𝐼 ‘ 𝐽 ) ∧ { 𝐵 , 𝐶 } ⊆ ( 𝐼 ‘ 𝐾 ) ) ) )
14	6 13	syl	⊢ ( 𝜑 → ( ( { 𝐴 , 𝐵 } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 0 ) ) ∧ { 𝐵 , 𝐶 } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 1 ) ) ) ↔ ( { 𝐴 , 𝐵 } ⊆ ( 𝐼 ‘ 𝐽 ) ∧ { 𝐵 , 𝐶 } ⊆ ( 𝐼 ‘ 𝐾 ) ) ) )
15	5 14	mpbird	⊢ ( 𝜑 → ( { 𝐴 , 𝐵 } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 0 ) ) ∧ { 𝐵 , 𝐶 } ⊆ ( 𝐼 ‘ ( 𝐹 ‘ 1 ) ) ) )