Theorem cicrcl2

Description: Isomorphism implies the structure being a category. (Contributed by Zhi Wang, 26-Oct-2025)

	Ref	Expression
Assertion	cicrcl2	⊢ ( 𝑅 ( ≃𝑐 ‘ 𝐶 ) 𝑆 → 𝐶 ∈ Cat )

Proof

Step	Hyp	Ref	Expression
1		df-br	⊢ ( 𝑅 ( ≃𝑐 ‘ 𝐶 ) 𝑆 ↔ ⟨ 𝑅 , 𝑆 ⟩ ∈ ( ≃𝑐 ‘ 𝐶 ) )
2		elfvdm	⊢ ( ⟨ 𝑅 , 𝑆 ⟩ ∈ ( ≃𝑐 ‘ 𝐶 ) → 𝐶 ∈ dom ≃𝑐 )
3		cicfn	⊢ ≃𝑐 Fn Cat
4	3	fndmi	⊢ dom ≃𝑐 = Cat
5	2 4	eleqtrdi	⊢ ( ⟨ 𝑅 , 𝑆 ⟩ ∈ ( ≃𝑐 ‘ 𝐶 ) → 𝐶 ∈ Cat )
6	1 5	sylbi	⊢ ( 𝑅 ( ≃𝑐 ‘ 𝐶 ) 𝑆 → 𝐶 ∈ Cat )