Theorem 3wlkdlem3

Description: Lemma 3 for 3wlkd . (Contributed by Alexander van der Vekens, 10-Nov-2017) (Revised 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 ) ) )
Assertion	3wlkdlem3	⊢ ( 𝜑 → ( ( ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ 1 ) = 𝐵 ) ∧ ( ( 𝑃 ‘ 2 ) = 𝐶 ∧ ( 𝑃 ‘ 3 ) = 𝐷 ) ) )

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	1	fveq1i	⊢ ( 𝑃 ‘ 0 ) = ( ⟨“ 𝐴 𝐵 𝐶 𝐷 ”⟩ ‘ 0 )
5		s4fv0	⊢ ( 𝐴 ∈ 𝑉 → ( ⟨“ 𝐴 𝐵 𝐶 𝐷 ”⟩ ‘ 0 ) = 𝐴 )
6	4 5	eqtrid	⊢ ( 𝐴 ∈ 𝑉 → ( 𝑃 ‘ 0 ) = 𝐴 )
7	1	fveq1i	⊢ ( 𝑃 ‘ 1 ) = ( ⟨“ 𝐴 𝐵 𝐶 𝐷 ”⟩ ‘ 1 )
8		s4fv1	⊢ ( 𝐵 ∈ 𝑉 → ( ⟨“ 𝐴 𝐵 𝐶 𝐷 ”⟩ ‘ 1 ) = 𝐵 )
9	7 8	eqtrid	⊢ ( 𝐵 ∈ 𝑉 → ( 𝑃 ‘ 1 ) = 𝐵 )
10	6 9	anim12i	⊢ ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) → ( ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ 1 ) = 𝐵 ) )
11	1	fveq1i	⊢ ( 𝑃 ‘ 2 ) = ( ⟨“ 𝐴 𝐵 𝐶 𝐷 ”⟩ ‘ 2 )
12		s4fv2	⊢ ( 𝐶 ∈ 𝑉 → ( ⟨“ 𝐴 𝐵 𝐶 𝐷 ”⟩ ‘ 2 ) = 𝐶 )
13	11 12	eqtrid	⊢ ( 𝐶 ∈ 𝑉 → ( 𝑃 ‘ 2 ) = 𝐶 )
14	1	fveq1i	⊢ ( 𝑃 ‘ 3 ) = ( ⟨“ 𝐴 𝐵 𝐶 𝐷 ”⟩ ‘ 3 )
15		s4fv3	⊢ ( 𝐷 ∈ 𝑉 → ( ⟨“ 𝐴 𝐵 𝐶 𝐷 ”⟩ ‘ 3 ) = 𝐷 )
16	14 15	eqtrid	⊢ ( 𝐷 ∈ 𝑉 → ( 𝑃 ‘ 3 ) = 𝐷 )
17	13 16	anim12i	⊢ ( ( 𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉 ) → ( ( 𝑃 ‘ 2 ) = 𝐶 ∧ ( 𝑃 ‘ 3 ) = 𝐷 ) )
18	10 17	anim12i	⊢ ( ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉 ) ) → ( ( ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ 1 ) = 𝐵 ) ∧ ( ( 𝑃 ‘ 2 ) = 𝐶 ∧ ( 𝑃 ‘ 3 ) = 𝐷 ) ) )
19	3 18	syl	⊢ ( 𝜑 → ( ( ( 𝑃 ‘ 0 ) = 𝐴 ∧ ( 𝑃 ‘ 1 ) = 𝐵 ) ∧ ( ( 𝑃 ‘ 2 ) = 𝐶 ∧ ( 𝑃 ‘ 3 ) = 𝐷 ) ) )