Theorem oppccat

Description: An opposite category is a category. (Contributed by Mario Carneiro, 2-Jan-2017)

	Ref	Expression
Hypothesis	oppcbas.1	$$⊢O=\mathrm{oppCat}\left(C\right)$$
Assertion	oppccat	$$⊢C\in \mathrm{Cat}\to O\in \mathrm{Cat}$$

Proof

Step	Hyp	Ref	Expression
1		oppcbas.1	$$⊢O=\mathrm{oppCat}\left(C\right)$$
2	1	oppccatid	$$⊢C\in \mathrm{Cat}\to \left(O\in \mathrm{Cat}\wedge \mathrm{Id}\left(O\right)=\mathrm{Id}\left(C\right)\right)$$
3	2	simpld	$$⊢C\in \mathrm{Cat}\to O\in \mathrm{Cat}$$