Theorem isorcl 49064

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 48879	. . . 4 ⊢ (𝐹 ∈ (𝐼‘⟨𝑋, 𝑌⟩) → 𝐼 ≠ ∅)
3		df-ov 7349	. . . 4 ⊢ (𝑋𝐼𝑌) = (𝐼‘⟨𝑋, 𝑌⟩)
4	2, 3	eleq2s 2849	. . 3 ⊢ (𝐹 ∈ (𝑋𝐼𝑌) → 𝐼 ≠ ∅)
5		isorcl.i	. . . . 5 ⊢ 𝐼 = (Iso‘𝐶)
6	5	neeq1i 2992	. . . 4 ⊢ (𝐼 ≠ ∅ ↔ (Iso‘𝐶) ≠ ∅)
7		n0 4303	. . . 4 ⊢ ((Iso‘𝐶) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (Iso‘𝐶))
8	6, 7	bitri 275	. . 3 ⊢ (𝐼 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (Iso‘𝐶))
9	4, 8	sylib 218	. 2 ⊢ (𝐹 ∈ (𝑋𝐼𝑌) → ∃𝑥 𝑥 ∈ (Iso‘𝐶))
10		df-iso 17653	. . . 4 ⊢ Iso = (𝑐 ∈ Cat ↦ ((𝑥 ∈ V ↦ dom 𝑥) ∘ (Inv‘𝑐)))
11	10	mptrcl 6938	. . 3 ⊢ (𝑥 ∈ (Iso‘𝐶) → 𝐶 ∈ Cat)
12	11	exlimiv 1931	. 2 ⊢ (∃𝑥 𝑥 ∈ (Iso‘𝐶) → 𝐶 ∈ Cat)
13	1, 9, 12	3syl 18	1 ⊢ (𝜑 → 𝐶 ∈ Cat)

Syntax hints:   → wi 4   = wceq 1541  ∃wex 1780   ∈ wcel 2111   ≠ wne 2928  Vcvv 3436  ∅c0 4283  ⟨cop 4582   ↦ cmpt 5172  dom cdm 5616   ∘ ccom 5620  ‘cfv 6481  (class class class)co 7346  Catccat 17567  Invcinv 17649  Isociso 17650

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pr 5370

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-rab 3396  df-v 3438  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-br 5092  df-opab 5154  df-mpt 5173  df-xp 5622  df-rel 5623  df-cnv 5624  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-iota 6437  df-fv 6489  df-ov 7349  df-iso 17653

This theorem is referenced by:  isorcl2  49065  isoval2  49066