Theorem oppcid

Description: Identity function of an opposite category. (Contributed by Mario Carneiro, 2-Jan-2017)

	Ref	Expression
Hypotheses	oppcbas.1	|- O = ( oppCat ` C )
	oppcid.2	|- B = ( Id ` C )
Assertion	oppcid	|- ( C e. Cat -> ( Id ` O ) = B )

Proof

Step	Hyp	Ref	Expression
1		oppcbas.1	|- O = ( oppCat ` C )
2		oppcid.2	|- B = ( Id ` C )
3	1	oppccatid	|- ( C e. Cat -> ( O e. Cat /\ ( Id ` O ) = ( Id ` C ) ) )
4	3	simprd	|- ( C e. Cat -> ( Id ` O ) = ( Id ` C ) )
5	4 2	eqtr4di	|- ( C e. Cat -> ( Id ` O ) = B )