Theorem isorcl 49788

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

Hypotheses

Ref	Expression
isorcl.i	⊢ 𝐼 = (Iso‘𝐶)
isorcl.f	⊢ (𝜑 → 𝐹 ∈ (𝑋𝐼𝑌))

Assertion

Ref	Expression
isorcl	⊢ (𝜑 → 𝐶 ∈ Cat)

Proof of Theorem isorcl

Dummy variables 𝑥 𝑐 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		isorcl.f	. 2 ⊢ (𝜑 → 𝐹 ∈ (𝑋𝐼𝑌))
2		elfvne0 49604	. . . 4 ⊢ (𝐹 ∈ (𝐼‘⟨𝑋, 𝑌⟩) → 𝐼 ≠ ∅)
3		df-ov 7415	. . . 4 ⊢ (𝑋𝐼𝑌) = (𝐼‘⟨𝑋, 𝑌⟩)
4	2, 3	eleq2s 2881	. . 3 ⊢ (𝐹 ∈ (𝑋𝐼𝑌) → 𝐼 ≠ ∅)
5		isorcl.i	. . . . 5 ⊢ 𝐼 = (Iso‘𝐶)
6	5	neeq1i 3022	. . . 4 ⊢ (𝐼 ≠ ∅ ↔ (Iso‘𝐶) ≠ ∅)
7		n0 4308	. . . 4 ⊢ ((Iso‘𝐶) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (Iso‘𝐶))
8	6, 7	bitri 278	. . 3 ⊢ (𝐼 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (Iso‘𝐶))
9	4, 8	sylib 221	. 2 ⊢ (𝐹 ∈ (𝑋𝐼𝑌) → ∃𝑥 𝑥 ∈ (Iso‘𝐶))
10		df-iso 17807	. . . 4 ⊢ Iso = (𝑐 ∈ Cat ↦ ((𝑥 ∈ V ↦ dom 𝑥) ∘ (Inv‘𝑐)))
11	10	mptrcl 7001	. . 3 ⊢ (𝑥 ∈ (Iso‘𝐶) → 𝐶 ∈ Cat)
12	11	exlimiv 1960	. 2 ⊢ (∃𝑥 𝑥 ∈ (Iso‘𝐶) → 𝐶 ∈ Cat)
13	1, 9, 12	3syl 19	1 ⊢ (𝜑 → 𝐶 ∈ Cat)

Syntax hints:   → wi 4   = wceq 1570  ∃wex 1809   ∈ wcel 2143   ≠ wne 2958  Vcvv 3455  ∅c0 4287  ⟨cop 4596   ↦ cmpt 5193  dom cdm 5663   ∘ ccom 5667  ‘cfv 6538  (class class class)co 7412  Catccat 17721  Invcinv 17803  Isociso 17804

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-nul 5270  ax-pr 5406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-opab 5175  df-mpt 5194  df-xp 5669  df-rel 5670  df-cnv 5671  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6494  df-fv 6546  df-ov 7415  df-iso 17807

This theorem is referenced by:  isorcl2  49789  isoval2  49790