Theorem oppctermhom

Description: The opposite category of a terminal category has the same base and hom-sets as the original category. (Contributed by Zhi Wang, 16-Oct-2025)

	Ref	Expression
Hypotheses	oppcterm.o	|- O = ( oppCat ` C )
	oppcterm.c	|- ( ph -> C e. TermCat )
Assertion	oppctermhom	|- ( ph -> ( Homf ` C ) = ( Homf ` O ) )

Proof

Step	Hyp	Ref	Expression
1		oppcterm.o	|- O = ( oppCat ` C )
2		oppcterm.c	|- ( ph -> C e. TermCat )
3		eqid	|- ( Base ` C ) = ( Base ` C )
4	2 3	termcbas	|- ( ph -> E. x ( Base ` C ) = { x } )
5		id	|- ( ( Base ` C ) = { x } -> ( Base ` C ) = { x } )
6	1 3 5	oppcmndc	|- ( ( Base ` C ) = { x } -> ( Homf ` C ) = ( Homf ` O ) )
7	6	exlimiv	|- ( E. x ( Base ` C ) = { x } -> ( Homf ` C ) = ( Homf ` O ) )
8	4 7	syl	|- ( ph -> ( Homf ` C ) = ( Homf ` O ) )