Theorem 2oppcbas

Description: The double opposite category has the same objects as the original category. Intended for use with property lemmas such as monpropd . (Contributed by Mario Carneiro, 3-Jan-2017)

	Ref	Expression
Hypotheses	oppcbas.1	$$⊢O=\mathrm{oppCat}\left(C\right)$$
	2oppcco.2	$$⊢B={\mathrm{Base}}_C$$
Assertion	2oppcbas	$$⊢B={\mathrm{Base}}_{\mathrm{oppCat}\left(O\right)}$$

Proof

Step	Hyp	Ref	Expression
1		oppcbas.1	$$⊢O=\mathrm{oppCat}\left(C\right)$$
2		2oppcco.2	$$⊢B={\mathrm{Base}}_C$$
3		eqid	$$⊢\mathrm{oppCat}\left(O\right)=\mathrm{oppCat}\left(O\right)$$
4	1 2	oppcbas	$$⊢B={\mathrm{Base}}_O$$
5	3 4	oppcbas	$$⊢B={\mathrm{Base}}_{\mathrm{oppCat}\left(O\right)}$$