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 ∈ 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 ∈ Cat → ( 𝑂 ∈ Cat ∧ ( Id ‘ 𝑂 ) = ( Id ‘ 𝐶 ) ) )
4	3	simprd	⊢ ( 𝐶 ∈ Cat → ( Id ‘ 𝑂 ) = ( Id ‘ 𝐶 ) )
5	4 2	eqtr4di	⊢ ( 𝐶 ∈ Cat → ( Id ‘ 𝑂 ) = 𝐵 )