Theorem 2wlkdlem3

Description: Lemma 3 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 ) )
Assertion	2wlkdlem3	⊢ ( 𝜑 → ( ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ 1 ) = 𝐵 ∧ ( 𝑃 ‘ 2 ) = 𝐶 ) )

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	1	fveq1i	⊢ ( 𝑃 ‘ 0 ) = ( ⟨“ 𝐴 𝐵 𝐶 ”⟩ ‘ 0 )
5		s3fv0	⊢ ( 𝐴 ∈ 𝑉 → ( ⟨“ 𝐴 𝐵 𝐶 ”⟩ ‘ 0 ) = 𝐴 )
6	4 5	eqtrid	⊢ ( 𝐴 ∈ 𝑉 → ( 𝑃 ‘ 0 ) = 𝐴 )
7	1	fveq1i	⊢ ( 𝑃 ‘ 1 ) = ( ⟨“ 𝐴 𝐵 𝐶 ”⟩ ‘ 1 )
8		s3fv1	⊢ ( 𝐵 ∈ 𝑉 → ( ⟨“ 𝐴 𝐵 𝐶 ”⟩ ‘ 1 ) = 𝐵 )
9	7 8	eqtrid	⊢ ( 𝐵 ∈ 𝑉 → ( 𝑃 ‘ 1 ) = 𝐵 )
10	1	fveq1i	⊢ ( 𝑃 ‘ 2 ) = ( ⟨“ 𝐴 𝐵 𝐶 ”⟩ ‘ 2 )
11		s3fv2	⊢ ( 𝐶 ∈ 𝑉 → ( ⟨“ 𝐴 𝐵 𝐶 ”⟩ ‘ 2 ) = 𝐶 )
12	10 11	eqtrid	⊢ ( 𝐶 ∈ 𝑉 → ( 𝑃 ‘ 2 ) = 𝐶 )
13	6 9 12	3anim123i	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉 ) → ( ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ 1 ) = 𝐵 ∧ ( 𝑃 ‘ 2 ) = 𝐶 ) )
14	3 13	syl	⊢ ( 𝜑 → ( ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ 1 ) = 𝐵 ∧ ( 𝑃 ‘ 2 ) = 𝐶 ) )