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Metamath Proof Explorer

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 ) )