Theorem oppcthinendc

Description: The opposite category of a thin category whose morphisms are all endomorphisms has the same base, hom-sets ( oppcendc ) and composition operation as the original category. (Contributed by Zhi Wang, 16-Oct-2025)

	Ref	Expression
Hypotheses	oppcthinco.o	⊢ 𝑂 = ( oppCat ‘ 𝐶 )
	oppcthinco.c	⊢ ( 𝜑 → 𝐶 ∈ ThinCat )
	oppcthinendc.b	⊢ 𝐵 = ( Base ‘ 𝐶 )
	oppcthinendc.h	⊢ 𝐻 = ( Hom ‘ 𝐶 )
	oppcthinendc.1	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝑥 ≠ 𝑦 → ( 𝑥 𝐻 𝑦 ) = ∅ ) )
Assertion	oppcthinendc	⊢ ( 𝜑 → ( compf ‘ 𝐶 ) = ( compf ‘ 𝑂 ) )

Proof

Step	Hyp	Ref	Expression
1		oppcthinco.o	⊢ 𝑂 = ( oppCat ‘ 𝐶 )
2		oppcthinco.c	⊢ ( 𝜑 → 𝐶 ∈ ThinCat )
3		oppcthinendc.b	⊢ 𝐵 = ( Base ‘ 𝐶 )
4		oppcthinendc.h	⊢ 𝐻 = ( Hom ‘ 𝐶 )
5		oppcthinendc.1	⊢ ( ( 𝜑 ∧ ( 𝑥 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵 ) ) → ( 𝑥 ≠ 𝑦 → ( 𝑥 𝐻 𝑦 ) = ∅ ) )
6	1 3 4 5	oppcendc	⊢ ( 𝜑 → ( Homf ‘ 𝐶 ) = ( Homf ‘ 𝑂 ) )
7	1 2 6	oppcthinco	⊢ ( 𝜑 → ( compf ‘ 𝐶 ) = ( compf ‘ 𝑂 ) )