Theorem elsetchom

Description: A morphism of sets is a function. (Contributed by Mario Carneiro, 3-Jan-2017)

	Ref	Expression
Hypotheses	setcbas.c	⊢ 𝐶 = ( SetCat ‘ 𝑈 )
	setcbas.u	⊢ ( 𝜑 → 𝑈 ∈ 𝑉 )
	setchomfval.h	⊢ 𝐻 = ( Hom ‘ 𝐶 )
	setchom.x	⊢ ( 𝜑 → 𝑋 ∈ 𝑈 )
	setchom.y	⊢ ( 𝜑 → 𝑌 ∈ 𝑈 )
Assertion	elsetchom	⊢ ( 𝜑 → ( 𝐹 ∈ ( 𝑋 𝐻 𝑌 ) ↔ 𝐹 : 𝑋 ⟶ 𝑌 ) )

Proof

Step	Hyp	Ref	Expression
1		setcbas.c	⊢ 𝐶 = ( SetCat ‘ 𝑈 )
2		setcbas.u	⊢ ( 𝜑 → 𝑈 ∈ 𝑉 )
3		setchomfval.h	⊢ 𝐻 = ( Hom ‘ 𝐶 )
4		setchom.x	⊢ ( 𝜑 → 𝑋 ∈ 𝑈 )
5		setchom.y	⊢ ( 𝜑 → 𝑌 ∈ 𝑈 )
6	1 2 3 4 5	setchom	⊢ ( 𝜑 → ( 𝑋 𝐻 𝑌 ) = ( 𝑌 ↑m 𝑋 ) )
7	6	eleq2d	⊢ ( 𝜑 → ( 𝐹 ∈ ( 𝑋 𝐻 𝑌 ) ↔ 𝐹 ∈ ( 𝑌 ↑m 𝑋 ) ) )
8	5 4	elmapd	⊢ ( 𝜑 → ( 𝐹 ∈ ( 𝑌 ↑m 𝑋 ) ↔ 𝐹 : 𝑋 ⟶ 𝑌 ) )
9	7 8	bitrd	⊢ ( 𝜑 → ( 𝐹 ∈ ( 𝑋 𝐻 𝑌 ) ↔ 𝐹 : 𝑋 ⟶ 𝑌 ) )