Theorem oppccat

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

	Ref	Expression
Hypothesis	oppcbas.1	⊢ 𝑂 = ( oppCat ‘ 𝐶 )
Assertion	oppccat	⊢ ( 𝐶 ∈ Cat → 𝑂 ∈ Cat )

Proof

Step	Hyp	Ref	Expression
1		oppcbas.1	⊢ 𝑂 = ( oppCat ‘ 𝐶 )
2	1	oppccatid	⊢ ( 𝐶 ∈ Cat → ( 𝑂 ∈ Cat ∧ ( Id ‘ 𝑂 ) = ( Id ‘ 𝐶 ) ) )
3	2	simpld	⊢ ( 𝐶 ∈ Cat → 𝑂 ∈ Cat )