Theorem setccat

Description: The category of sets is a category. (Contributed by Mario Carneiro, 3-Jan-2017)

	Ref	Expression
Hypothesis	setccat.c	⊢ 𝐶 = ( SetCat ‘ 𝑈 )
Assertion	setccat	⊢ ( 𝑈 ∈ 𝑉 → 𝐶 ∈ Cat )

Proof

Step	Hyp	Ref	Expression
1		setccat.c	⊢ 𝐶 = ( SetCat ‘ 𝑈 )
2	1	setccatid	⊢ ( 𝑈 ∈ 𝑉 → ( 𝐶 ∈ Cat ∧ ( Id ‘ 𝐶 ) = ( 𝑥 ∈ 𝑈 ↦ ( I ↾ 𝑥 ) ) ) )
3	2	simpld	⊢ ( 𝑈 ∈ 𝑉 → 𝐶 ∈ Cat )