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 ) ) ) )