Theorem oppctermco

Description: The opposite category of a terminal category has the same base, hom-sets and composition operation as the original category. Note that C = O cannot be proved because C might not even be a function. For example, let C be ( { <. ( Basendx ) , { (/) } >. , <. ( Homndx ) , ( (V X. V ) X. { { (/) } } ) >. } u. { <. ( compndx ) , { (/) } >. , <. ( compndx ) , 2o >. } ) ; it should be a terminal category, but the opposite category is not itself. See the definitions df-oppc and df-sets . (Contributed by Zhi Wang, 16-Oct-2025)

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

Proof

Step	Hyp	Ref	Expression
1		oppcterm.o	|- O = ( oppCat ` C )
2		oppcterm.c	|- ( ph -> C e. TermCat )
3	2	termcthind	|- ( ph -> C e. ThinCat )
4	1 2	oppctermhom	|- ( ph -> ( Homf ` C ) = ( Homf ` O ) )
5	1 3 4	oppcthinco	|- ( ph -> ( comf ` C ) = ( comf ` O ) )