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	⊢ ( 𝜑 → 𝐶 ∈ TermCat )
Assertion	oppctermhom	⊢ ( 𝜑 → ( Homf ‘ 𝐶 ) = ( Homf ‘ 𝑂 ) )

Proof

Step	Hyp	Ref	Expression
1		oppcterm.o	⊢ 𝑂 = ( oppCat ‘ 𝐶 )
2		oppcterm.c	⊢ ( 𝜑 → 𝐶 ∈ TermCat )
3		eqid	⊢ ( Base ‘ 𝐶 ) = ( Base ‘ 𝐶 )
4	2 3	termcbas	⊢ ( 𝜑 → ∃ 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	⊢ ( ∃ 𝑥 ( Base ‘ 𝐶 ) = { 𝑥 } → ( Homf ‘ 𝐶 ) = ( Homf ‘ 𝑂 ) )
8	4 7	syl	⊢ ( 𝜑 → ( Homf ‘ 𝐶 ) = ( Homf ‘ 𝑂 ) )