Theorem setchom

Description: Set of arrows of the category of sets (in a universe). (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	setchom	⊢ ( 𝜑 → ( 𝑋 𝐻 𝑌 ) = ( 𝑌 ↑m 𝑋 ) )

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	setchomfval	⊢ ( 𝜑 → 𝐻 = ( 𝑥 ∈ 𝑈 , 𝑦 ∈ 𝑈 ↦ ( 𝑦 ↑m 𝑥 ) ) )
7		simprr	⊢ ( ( 𝜑 ∧ ( 𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ) ) → 𝑦 = 𝑌 )
8		simprl	⊢ ( ( 𝜑 ∧ ( 𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ) ) → 𝑥 = 𝑋 )
9	7 8	oveq12d	⊢ ( ( 𝜑 ∧ ( 𝑥 = 𝑋 ∧ 𝑦 = 𝑌 ) ) → ( 𝑦 ↑m 𝑥 ) = ( 𝑌 ↑m 𝑋 ) )
10		ovexd	⊢ ( 𝜑 → ( 𝑌 ↑m 𝑋 ) ∈ V )
11	6 9 4 5 10	ovmpod	⊢ ( 𝜑 → ( 𝑋 𝐻 𝑌 ) = ( 𝑌 ↑m 𝑋 ) )