Theorem sectrcl 50341

Description: Reverse closure for section relations. (Contributed by Zhi Wang, 14-Nov-2025.)

Hypotheses

Ref	Expression
sectrcl.s	⊢ 𝑆 = (Sect‘𝐶)
sectrcl.f	⊢ (𝜑 → 𝐹(𝑋𝑆𝑌)𝐺)

Assertion

Ref	Expression
sectrcl	⊢ (𝜑 → 𝐶 ∈ Cat)

Proof of Theorem sectrcl

Dummy variables 𝑥 𝑦 𝑐 𝑓 𝑔 ℎ are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		sectrcl.f	. 2 ⊢ (𝜑 → 𝐹(𝑋𝑆𝑌)𝐺)
2		df-br 5107	. . . . 5 ⊢ (𝐹(𝑋𝑆𝑌)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝑋𝑆𝑌))
3		df-ov 7446	. . . . . 6 ⊢ (𝑋𝑆𝑌) = (𝑆‘⟨𝑋, 𝑌⟩)
4	3	eleq2i 2856	. . . . 5 ⊢ (⟨𝐹, 𝐺⟩ ∈ (𝑋𝑆𝑌) ↔ ⟨𝐹, 𝐺⟩ ∈ (𝑆‘⟨𝑋, 𝑌⟩))
5	2, 4	bitri 278	. . . 4 ⊢ (𝐹(𝑋𝑆𝑌)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝑆‘⟨𝑋, 𝑌⟩))
6		elfvne0 50165	. . . 4 ⊢ (⟨𝐹, 𝐺⟩ ∈ (𝑆‘⟨𝑋, 𝑌⟩) → 𝑆 ≠ ∅)
7	5, 6	sylbi 220	. . 3 ⊢ (𝐹(𝑋𝑆𝑌)𝐺 → 𝑆 ≠ ∅)
8		sectrcl.s	. . . . 5 ⊢ 𝑆 = (Sect‘𝐶)
9	8	neeq1i 3023	. . . 4 ⊢ (𝑆 ≠ ∅ ↔ (Sect‘𝐶) ≠ ∅)
10		n0 4303	. . . 4 ⊢ ((Sect‘𝐶) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (Sect‘𝐶))
11	9, 10	bitri 278	. . 3 ⊢ (𝑆 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (Sect‘𝐶))
12	7, 11	sylib 221	. 2 ⊢ (𝐹(𝑋𝑆𝑌)𝐺 → ∃𝑥 𝑥 ∈ (Sect‘𝐶))
13		df-sect 17951	. . . 4 ⊢ Sect = (𝑐 ∈ Cat ↦ (𝑥 ∈ (Base‘𝑐), 𝑦 ∈ (Base‘𝑐) ↦ {⟨𝑓, 𝑔⟩ ∣ [(Hom ‘𝑐) / ℎ]((𝑓 ∈ (𝑥ℎ𝑦) ∧ 𝑔 ∈ (𝑦ℎ𝑥)) ∧ (𝑔(⟨𝑥, 𝑦⟩(comp‘𝑐)𝑥)𝑓) = ((Id‘𝑐)‘𝑥))}))
14	13	mptrcl 7026	. . 3 ⊢ (𝑥 ∈ (Sect‘𝐶) → 𝐶 ∈ Cat)
15	14	exlimiv 1965	. 2 ⊢ (∃𝑥 𝑥 ∈ (Sect‘𝐶) → 𝐶 ∈ Cat)
16	1, 12, 15	3syl 19	1 ⊢ (𝜑 → 𝐶 ∈ Cat)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2147   ≠ wne 2959  [wsbc 3742  ∅c0 4281  ⟨cop 4838   class class class wbr 5106  {copab 5170  ‘cfv 6558  (class class class)co 7443   ∈ cmpo 7445  Basecbs 17416  Hom chom 17468  compcco 17469  Catccat 17867  Idccid 17868  Sectcsect 17948

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 5252  ax-nul 5263  ax-pr 5395

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 5171  df-mpt 5190  df-xp 5661  df-rel 5662  df-cnv 5663  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-iota 6512  df-fv 6566  df-ov 7446  df-sect 17951

This theorem is used by:  sectrcl2  50342  isinv2  50345  catcsect  50738