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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 → ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉 ) ) )
3wlkd.n ⊢ ( 𝜑 → ( ( 𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ) ∧ ( 𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷 ) ∧ 𝐶 ≠ 𝐷 ) )
3wlkd.e ⊢ ( 𝜑 → ( { 𝐴 , 𝐵 } ⊆ ( 𝐼 ‘ 𝐽 ) ∧ { 𝐵 , 𝐶 } ⊆ ( 𝐼 ‘ 𝐾 ) ∧ { 𝐶 , 𝐷 } ⊆ ( 𝐼 ‘ 𝐿 ) ) )
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 → ( ( 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ) ∧ ( 𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑉 ) ) )
4 3wlkd.n ⊢ ( 𝜑 → ( ( 𝐴 ≠ 𝐵 ∧ 𝐴 ≠ 𝐶 ) ∧ ( 𝐵 ≠ 𝐶 ∧ 𝐵 ≠ 𝐷 ) ∧ 𝐶 ≠ 𝐷 ) )
5 3wlkd.e ⊢ ( 𝜑 → ( { 𝐴 , 𝐵 } ⊆ ( 𝐼 ‘ 𝐽 ) ∧ { 𝐵 , 𝐶 } ⊆ ( 𝐼 ‘ 𝐾 ) ∧ { 𝐶 , 𝐷 } ⊆ ( 𝐼 ‘ 𝐿 ) ) )
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 ) )