Theorem setccat

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

	Ref	Expression
Hypothesis	setccat.c	$$⊢C=\mathrm{SetCat}\left(U\right)$$
Assertion	setccat	$$⊢U\in V\to C\in \mathrm{Cat}$$

Proof

Step	Hyp	Ref	Expression
1		setccat.c	$$⊢C=\mathrm{SetCat}\left(U\right)$$
2	1	setccatid	$$⊢U\in V\to \left(C\in \mathrm{Cat}\wedge \mathrm{Id}\left(C\right)=\left(x\in U⟼{\mathrm{I}\upharpoonright }_x\right)\right)$$
3	2	simpld	$$⊢U\in V\to C\in \mathrm{Cat}$$