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	⊢ ( 𝜑 → 𝐶 ∈ TermCat )
Assertion	oppctermco	⊢ ( 𝜑 → ( compf ‘ 𝐶 ) = ( compf ‘ 𝑂 ) )

Proof

Step	Hyp	Ref	Expression
1		oppcterm.o	⊢ 𝑂 = ( oppCat ‘ 𝐶 )
2		oppcterm.c	⊢ ( 𝜑 → 𝐶 ∈ TermCat )
3	2	termcthind	⊢ ( 𝜑 → 𝐶 ∈ ThinCat )
4	1 2	oppctermhom	⊢ ( 𝜑 → ( Homf ‘ 𝐶 ) = ( Homf ‘ 𝑂 ) )
5	1 3 4	oppcthinco	⊢ ( 𝜑 → ( compf ‘ 𝐶 ) = ( compf ‘ 𝑂 ) )