Metamath Proof Explorer


Theorem 3wlkdlem8

Description: Lemma 8 for 3wlkd . (Contributed by Alexander van der Vekens, 12-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 → ( ( 𝐴𝑉𝐵𝑉 ) ∧ ( 𝐶𝑉𝐷𝑉 ) ) )
3wlkd.n ( 𝜑 → ( ( 𝐴𝐵𝐴𝐶 ) ∧ ( 𝐵𝐶𝐵𝐷 ) ∧ 𝐶𝐷 ) )
3wlkd.e ( 𝜑 → ( { 𝐴 , 𝐵 } ⊆ ( 𝐼𝐽 ) ∧ { 𝐵 , 𝐶 } ⊆ ( 𝐼𝐾 ) ∧ { 𝐶 , 𝐷 } ⊆ ( 𝐼𝐿 ) ) )
Assertion 3wlkdlem8 ( 𝜑 → ( ( 𝐹 ‘ 0 ) = 𝐽 ∧ ( 𝐹 ‘ 1 ) = 𝐾 ∧ ( 𝐹 ‘ 2 ) = 𝐿 ) )

Proof

Step Hyp Ref Expression
1 3wlkd.p 𝑃 = ⟨“ 𝐴 𝐵 𝐶 𝐷 ”⟩
2 3wlkd.f 𝐹 = ⟨“ 𝐽 𝐾 𝐿 ”⟩
3 3wlkd.s ( 𝜑 → ( ( 𝐴𝑉𝐵𝑉 ) ∧ ( 𝐶𝑉𝐷𝑉 ) ) )
4 3wlkd.n ( 𝜑 → ( ( 𝐴𝐵𝐴𝐶 ) ∧ ( 𝐵𝐶𝐵𝐷 ) ∧ 𝐶𝐷 ) )
5 3wlkd.e ( 𝜑 → ( { 𝐴 , 𝐵 } ⊆ ( 𝐼𝐽 ) ∧ { 𝐵 , 𝐶 } ⊆ ( 𝐼𝐾 ) ∧ { 𝐶 , 𝐷 } ⊆ ( 𝐼𝐿 ) ) )
6 1 2 3 4 5 3wlkdlem7 ( 𝜑 → ( 𝐽 ∈ V ∧ 𝐾 ∈ V ∧ 𝐿 ∈ V ) )
7 s3fv0 ( 𝐽 ∈ V → ( ⟨“ 𝐽 𝐾 𝐿 ”⟩ ‘ 0 ) = 𝐽 )
8 s3fv1 ( 𝐾 ∈ V → ( ⟨“ 𝐽 𝐾 𝐿 ”⟩ ‘ 1 ) = 𝐾 )
9 s3fv2 ( 𝐿 ∈ V → ( ⟨“ 𝐽 𝐾 𝐿 ”⟩ ‘ 2 ) = 𝐿 )
10 7 8 9 3anim123i ( ( 𝐽 ∈ V ∧ 𝐾 ∈ V ∧ 𝐿 ∈ V ) → ( ( ⟨“ 𝐽 𝐾 𝐿 ”⟩ ‘ 0 ) = 𝐽 ∧ ( ⟨“ 𝐽 𝐾 𝐿 ”⟩ ‘ 1 ) = 𝐾 ∧ ( ⟨“ 𝐽 𝐾 𝐿 ”⟩ ‘ 2 ) = 𝐿 ) )
11 6 10 syl ( 𝜑 → ( ( ⟨“ 𝐽 𝐾 𝐿 ”⟩ ‘ 0 ) = 𝐽 ∧ ( ⟨“ 𝐽 𝐾 𝐿 ”⟩ ‘ 1 ) = 𝐾 ∧ ( ⟨“ 𝐽 𝐾 𝐿 ”⟩ ‘ 2 ) = 𝐿 ) )
12 2 fveq1i ( 𝐹 ‘ 0 ) = ( ⟨“ 𝐽 𝐾 𝐿 ”⟩ ‘ 0 )
13 12 eqeq1i ( ( 𝐹 ‘ 0 ) = 𝐽 ↔ ( ⟨“ 𝐽 𝐾 𝐿 ”⟩ ‘ 0 ) = 𝐽 )
14 2 fveq1i ( 𝐹 ‘ 1 ) = ( ⟨“ 𝐽 𝐾 𝐿 ”⟩ ‘ 1 )
15 14 eqeq1i ( ( 𝐹 ‘ 1 ) = 𝐾 ↔ ( ⟨“ 𝐽 𝐾 𝐿 ”⟩ ‘ 1 ) = 𝐾 )
16 2 fveq1i ( 𝐹 ‘ 2 ) = ( ⟨“ 𝐽 𝐾 𝐿 ”⟩ ‘ 2 )
17 16 eqeq1i ( ( 𝐹 ‘ 2 ) = 𝐿 ↔ ( ⟨“ 𝐽 𝐾 𝐿 ”⟩ ‘ 2 ) = 𝐿 )
18 13 15 17 3anbi123i ( ( ( 𝐹 ‘ 0 ) = 𝐽 ∧ ( 𝐹 ‘ 1 ) = 𝐾 ∧ ( 𝐹 ‘ 2 ) = 𝐿 ) ↔ ( ( ⟨“ 𝐽 𝐾 𝐿 ”⟩ ‘ 0 ) = 𝐽 ∧ ( ⟨“ 𝐽 𝐾 𝐿 ”⟩ ‘ 1 ) = 𝐾 ∧ ( ⟨“ 𝐽 𝐾 𝐿 ”⟩ ‘ 2 ) = 𝐿 ) )
19 11 18 sylibr ( 𝜑 → ( ( 𝐹 ‘ 0 ) = 𝐽 ∧ ( 𝐹 ‘ 1 ) = 𝐾 ∧ ( 𝐹 ‘ 2 ) = 𝐿 ) )