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Metamath Proof Explorer

Theorem elhoma

Description: Value of the disjointified hom-set function. (Contributed by Mario Carneiro, 11-Jan-2017)

Ref Expression
Hypotheses homarcl.h ⊢ 𝐻 = ( Homa ‘ 𝐶 )
homafval.b ⊢ 𝐵 = ( Base ‘ 𝐶 )
homafval.c ⊢ ( 𝜑 → 𝐶 ∈ Cat )
homaval.j ⊢ 𝐽 = ( Hom ‘ 𝐶 )
homaval.x ⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
homaval.y ⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
Assertion elhoma ( 𝜑 → ( 𝑍 ( 𝑋 𝐻 𝑌 ) 𝐹 ↔ ( 𝑍 = ⟨ 𝑋 , 𝑌 ⟩ ∧ 𝐹 ∈ ( 𝑋 𝐽 𝑌 ) ) ) )

Proof

Step Hyp Ref Expression
1 homarcl.h ⊢ 𝐻 = ( Homa ‘ 𝐶 )
2 homafval.b ⊢ 𝐵 = ( Base ‘ 𝐶 )
3 homafval.c ⊢ ( 𝜑 → 𝐶 ∈ Cat )
4 homaval.j ⊢ 𝐽 = ( Hom ‘ 𝐶 )
5 homaval.x ⊢ ( 𝜑 → 𝑋 ∈ 𝐵 )
6 homaval.y ⊢ ( 𝜑 → 𝑌 ∈ 𝐵 )
7 1 2 3 4 5 6 homaval ⊢ ( 𝜑 → ( 𝑋 𝐻 𝑌 ) = ( { ⟨ 𝑋 , 𝑌 ⟩ } × ( 𝑋 𝐽 𝑌 ) ) )
8 7 breqd ⊢ ( 𝜑 → ( 𝑍 ( 𝑋 𝐻 𝑌 ) 𝐹 ↔ 𝑍 ( { ⟨ 𝑋 , 𝑌 ⟩ } × ( 𝑋 𝐽 𝑌 ) ) 𝐹 ) )
9 brxp ⊢ ( 𝑍 ( { ⟨ 𝑋 , 𝑌 ⟩ } × ( 𝑋 𝐽 𝑌 ) ) 𝐹 ↔ ( 𝑍 ∈ { ⟨ 𝑋 , 𝑌 ⟩ } ∧ 𝐹 ∈ ( 𝑋 𝐽 𝑌 ) ) )
10 opex ⊢ ⟨ 𝑋 , 𝑌 ⟩ ∈ V
11 10 elsn2 ⊢ ( 𝑍 ∈ { ⟨ 𝑋 , 𝑌 ⟩ } ↔ 𝑍 = ⟨ 𝑋 , 𝑌 ⟩ )
12 11 anbi1i ⊢ ( ( 𝑍 ∈ { ⟨ 𝑋 , 𝑌 ⟩ } ∧ 𝐹 ∈ ( 𝑋 𝐽 𝑌 ) ) ↔ ( 𝑍 = ⟨ 𝑋 , 𝑌 ⟩ ∧ 𝐹 ∈ ( 𝑋 𝐽 𝑌 ) ) )
13 9 12 bitri ⊢ ( 𝑍 ( { ⟨ 𝑋 , 𝑌 ⟩ } × ( 𝑋 𝐽 𝑌 ) ) 𝐹 ↔ ( 𝑍 = ⟨ 𝑋 , 𝑌 ⟩ ∧ 𝐹 ∈ ( 𝑋 𝐽 𝑌 ) ) )
14 8 13 bitrdi ⊢ ( 𝜑 → ( 𝑍 ( 𝑋 𝐻 𝑌 ) 𝐹 ↔ ( 𝑍 = ⟨ 𝑋 , 𝑌 ⟩ ∧ 𝐹 ∈ ( 𝑋 𝐽 𝑌 ) ) ) )