Theorem 2wlkdlem8

Description: Lemma 8 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	2wlkdlem8	⊢ ( 𝜑 → ( ( 𝐹 ‘ 0 ) = 𝐽 ∧ ( 𝐹 ‘ 1 ) = 𝐾 ) )

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	2wlkdlem7	⊢ ( 𝜑 → ( 𝐽 ∈ V ∧ 𝐾 ∈ V ) )
7		s2fv0	⊢ ( 𝐽 ∈ V → ( ⟨“ 𝐽 𝐾 ”⟩ ‘ 0 ) = 𝐽 )
8		s2fv1	⊢ ( 𝐾 ∈ V → ( ⟨“ 𝐽 𝐾 ”⟩ ‘ 1 ) = 𝐾 )
9	7 8	anim12i	⊢ ( ( 𝐽 ∈ V ∧ 𝐾 ∈ V ) → ( ( ⟨“ 𝐽 𝐾 ”⟩ ‘ 0 ) = 𝐽 ∧ ( ⟨“ 𝐽 𝐾 ”⟩ ‘ 1 ) = 𝐾 ) )
10	6 9	syl	⊢ ( 𝜑 → ( ( ⟨“ 𝐽 𝐾 ”⟩ ‘ 0 ) = 𝐽 ∧ ( ⟨“ 𝐽 𝐾 ”⟩ ‘ 1 ) = 𝐾 ) )
11	2	fveq1i	⊢ ( 𝐹 ‘ 0 ) = ( ⟨“ 𝐽 𝐾 ”⟩ ‘ 0 )
12	11	eqeq1i	⊢ ( ( 𝐹 ‘ 0 ) = 𝐽 ↔ ( ⟨“ 𝐽 𝐾 ”⟩ ‘ 0 ) = 𝐽 )
13	2	fveq1i	⊢ ( 𝐹 ‘ 1 ) = ( ⟨“ 𝐽 𝐾 ”⟩ ‘ 1 )
14	13	eqeq1i	⊢ ( ( 𝐹 ‘ 1 ) = 𝐾 ↔ ( ⟨“ 𝐽 𝐾 ”⟩ ‘ 1 ) = 𝐾 )
15	12 14	anbi12i	⊢ ( ( ( 𝐹 ‘ 0 ) = 𝐽 ∧ ( 𝐹 ‘ 1 ) = 𝐾 ) ↔ ( ( ⟨“ 𝐽 𝐾 ”⟩ ‘ 0 ) = 𝐽 ∧ ( ⟨“ 𝐽 𝐾 ”⟩ ‘ 1 ) = 𝐾 ) )
16	10 15	sylibr	⊢ ( 𝜑 → ( ( 𝐹 ‘ 0 ) = 𝐽 ∧ ( 𝐹 ‘ 1 ) = 𝐾 ) )