Theorem isorcl 50362

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 50175	. . . 4 ⊢ (𝐹 ∈ (𝐼‘⟨𝑋, 𝑌⟩) → 𝐼 ≠ ∅)
3		df-ov 7449	. . . 4 ⊢ (𝑋𝐼𝑌) = (𝐼‘⟨𝑋, 𝑌⟩)
4	2, 3	eleq2s 2882	. . 3 ⊢ (𝐹 ∈ (𝑋𝐼𝑌) → 𝐼 ≠ ∅)
5		isorcl.i	. . . . 5 ⊢ 𝐼 = (Iso‘𝐶)
6	5	neeq1i 3023	. . . 4 ⊢ (𝐼 ≠ ∅ ↔ (Iso‘𝐶) ≠ ∅)
7		n0 4303	. . . 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 17956	. . . 4 ⊢ Iso = (𝑐 ∈ Cat ↦ ((𝑥 ∈ V ↦ dom 𝑥) ∘ (Inv‘𝑐)))
11	10	mptrcl 7029	. . 3 ⊢ (𝑥 ∈ (Iso‘𝐶) → 𝐶 ∈ Cat)
12	11	exlimiv 1965	. 2 ⊢ (∃𝑥 𝑥 ∈ (Iso‘𝐶) → 𝐶 ∈ Cat)
13	1, 9, 12	3syl 19	1 ⊢ (𝜑 → 𝐶 ∈ Cat)

This proof depends on syntax axioms:   → wi 4   = wceq 1570  ∃wex 1812   ∈ wcel 2147   ≠ wne 2959  Vcvv 3454  ∅c0 4281  ⟨cop 4838   ↦ cmpt 5190  dom cdm 5656   ∘ ccom 5660  ‘cfv 6561  (class class class)co 7446  Catccat 17870  Invcinv 17952  Isociso 17953

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5253  ax-nul 5264  ax-pr 5396

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5172  df-mpt 5191  df-xp 5662  df-rel 5663  df-cnv 5664  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-iota 6515  df-fv 6569  df-ov 7449  df-iso 17956

This theorem is used by:  isorcl2  50363  isoval2  50364