Theorem isorcl

Description: Reverse closure for isomorphism relations. (Contributed by Zhi Wang, 17-Nov-2025)

	Ref	Expression
Hypotheses	isorcl.i	⊢ 𝐼 = ( Iso ‘ 𝐶 )
	isorcl.f	⊢ ( 𝜑 → 𝐹 ∈ ( 𝑋 𝐼 𝑌 ) )
Assertion	isorcl	⊢ ( 𝜑 → 𝐶 ∈ Cat )

Proof

Step	Hyp	Ref	Expression
1		isorcl.i	⊢ 𝐼 = ( Iso ‘ 𝐶 )
2		isorcl.f	⊢ ( 𝜑 → 𝐹 ∈ ( 𝑋 𝐼 𝑌 ) )
3		elfvne0	⊢ ( 𝐹 ∈ ( 𝐼 ‘ ⟨ 𝑋 , 𝑌 ⟩ ) → 𝐼 ≠ ∅ )
4		df-ov	⊢ ( 𝑋 𝐼 𝑌 ) = ( 𝐼 ‘ ⟨ 𝑋 , 𝑌 ⟩ )
5	3 4	eleq2s	⊢ ( 𝐹 ∈ ( 𝑋 𝐼 𝑌 ) → 𝐼 ≠ ∅ )
6	1	neeq1i	⊢ ( 𝐼 ≠ ∅ ↔ ( Iso ‘ 𝐶 ) ≠ ∅ )
7		n0	⊢ ( ( Iso ‘ 𝐶 ) ≠ ∅ ↔ ∃ 𝑥 𝑥 ∈ ( Iso ‘ 𝐶 ) )
8	6 7	bitri	⊢ ( 𝐼 ≠ ∅ ↔ ∃ 𝑥 𝑥 ∈ ( Iso ‘ 𝐶 ) )
9	5 8	sylib	⊢ ( 𝐹 ∈ ( 𝑋 𝐼 𝑌 ) → ∃ 𝑥 𝑥 ∈ ( Iso ‘ 𝐶 ) )
10		df-iso	⊢ Iso = ( 𝑐 ∈ Cat ↦ ( ( 𝑥 ∈ V ↦ dom 𝑥 ) ∘ ( Inv ‘ 𝑐 ) ) )
11	10	mptrcl	⊢ ( 𝑥 ∈ ( Iso ‘ 𝐶 ) → 𝐶 ∈ Cat )
12	11	exlimiv	⊢ ( ∃ 𝑥 𝑥 ∈ ( Iso ‘ 𝐶 ) → 𝐶 ∈ Cat )
13	2 9 12	3syl	⊢ ( 𝜑 → 𝐶 ∈ Cat )