Theorem isorcl 49530

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 49346	. . . 4 ⊢ (𝐹 ∈ (𝐼‘⟨𝑋, 𝑌⟩) → 𝐼 ≠ ∅)
3		df-ov 7366	. . . 4 ⊢ (𝑋𝐼𝑌) = (𝐼‘⟨𝑋, 𝑌⟩)
4	2, 3	eleq2s 2858	. . 3 ⊢ (𝐹 ∈ (𝑋𝐼𝑌) → 𝐼 ≠ ∅)
5		isorcl.i	. . . . 5 ⊢ 𝐼 = (Iso‘𝐶)
6	5	neeq1i 2999	. . . 4 ⊢ (𝐼 ≠ ∅ ↔ (Iso‘𝐶) ≠ ∅)
7		n0 4288	. . . 4 ⊢ ((Iso‘𝐶) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (Iso‘𝐶))
8	6, 7	bitri 276	. . 3 ⊢ (𝐼 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (Iso‘𝐶))
9	4, 8	sylib 219	. 2 ⊢ (𝐹 ∈ (𝑋𝐼𝑌) → ∃𝑥 𝑥 ∈ (Iso‘𝐶))
10		df-iso 17714	. . . 4 ⊢ Iso = (𝑐 ∈ Cat ↦ ((𝑥 ∈ V ↦ dom 𝑥) ∘ (Inv‘𝑐)))
11	10	mptrcl 6952	. . 3 ⊢ (𝑥 ∈ (Iso‘𝐶) → 𝐶 ∈ Cat)
12	11	exlimiv 1937	. 2 ⊢ (∃𝑥 𝑥 ∈ (Iso‘𝐶) → 𝐶 ∈ Cat)
13	1, 9, 12	3syl 18	1 ⊢ (𝜑 → 𝐶 ∈ Cat)

Syntax hints:   → wi 4   = wceq 1547  ∃wex 1786   ∈ wcel 2119   ≠ wne 2935  Vcvv 3432  ∅c0 4268  ⟨cop 4568   ↦ cmpt 5160  dom cdm 5625   ∘ ccom 5629  ‘cfv 6492  (class class class)co 7363  Catccat 17628  Invcinv 17710  Isociso 17711

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-nul 5235  ax-pr 5369

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-br 5080  df-opab 5142  df-mpt 5161  df-xp 5631  df-rel 5632  df-cnv 5633  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-iota 6448  df-fv 6500  df-ov 7366  df-iso 17714

This theorem is referenced by:  isorcl2  49531  isoval2  49532