Theorem catccat

Description: The category of categories is a category, see remark 3.48 in Adamek p. 40. (Clearly it cannot be an element of itself, hence it is " U -large".) (Contributed by Mario Carneiro, 3-Jan-2017)

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

Proof

Step	Hyp	Ref	Expression
1		catccat.c	$$⊢C=\mathrm{CatCat}\left(U\right)$$
2		eqid	$$⊢{\mathrm{Base}}_C={\mathrm{Base}}_C$$
3	1 2	catccatid	$$⊢U\in V\to \left(C\in \mathrm{Cat}\wedge \mathrm{Id}\left(C\right)=\left(x\in {\mathrm{Base}}_C⟼{id}_{\mathrm{func}}\left(x\right)\right)\right)$$
4	3	simpld	$$⊢U\in V\to C\in \mathrm{Cat}$$