Theorem oppcmndc

Description: The opposite category of a category whose base set is a singleton or an empty set has the same base and hom-sets as the original category. (Contributed by Zhi Wang, 16-Oct-2025)

	Ref	Expression
Hypotheses	oppcendc.o	|- O = ( oppCat ` C )
	oppcendc.b	|- B = ( Base ` C )
	oppcmndc.x	|- ( ph -> B = { X } )
Assertion	oppcmndc	|- ( ph -> ( Homf ` C ) = ( Homf ` O ) )

Proof

Step	Hyp	Ref	Expression
1		oppcendc.o	|- O = ( oppCat ` C )
2		oppcendc.b	|- B = ( Base ` C )
3		oppcmndc.x	|- ( ph -> B = { X } )
4		eqid	|- ( Hom ` C ) = ( Hom ` C )
5	3	oppcmndclem	|- ( ( ph /\ ( x e. B /\ y e. B ) ) -> ( x =/= y -> ( x ( Hom ` C ) y ) = (/) ) )
6	1 2 4 5	oppcendc	|- ( ph -> ( Homf ` C ) = ( Homf ` O ) )