Theorem sectrcl 49513

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 5080	. . . . 5 ⊢ (𝐹(𝑋𝑆𝑌)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝑋𝑆𝑌))
3		df-ov 7366	. . . . . 6 ⊢ (𝑋𝑆𝑌) = (𝑆‘⟨𝑋, 𝑌⟩)
4	3	eleq2i 2832	. . . . 5 ⊢ (⟨𝐹, 𝐺⟩ ∈ (𝑋𝑆𝑌) ↔ ⟨𝐹, 𝐺⟩ ∈ (𝑆‘⟨𝑋, 𝑌⟩))
5	2, 4	bitri 276	. . . 4 ⊢ (𝐹(𝑋𝑆𝑌)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝑆‘⟨𝑋, 𝑌⟩))
6		elfvne0 49340	. . . 4 ⊢ (⟨𝐹, 𝐺⟩ ∈ (𝑆‘⟨𝑋, 𝑌⟩) → 𝑆 ≠ ∅)
7	5, 6	sylbi 218	. . 3 ⊢ (𝐹(𝑋𝑆𝑌)𝐺 → 𝑆 ≠ ∅)
8		sectrcl.s	. . . . 5 ⊢ 𝑆 = (Sect‘𝐶)
9	8	neeq1i 2999	. . . 4 ⊢ (𝑆 ≠ ∅ ↔ (Sect‘𝐶) ≠ ∅)
10		n0 4288	. . . 4 ⊢ ((Sect‘𝐶) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (Sect‘𝐶))
11	9, 10	bitri 276	. . 3 ⊢ (𝑆 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (Sect‘𝐶))
12	7, 11	sylib 219	. 2 ⊢ (𝐹(𝑋𝑆𝑌)𝐺 → ∃𝑥 𝑥 ∈ (Sect‘𝐶))
13		df-sect 17712	. . . 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 6952	. . 3 ⊢ (𝑥 ∈ (Sect‘𝐶) → 𝐶 ∈ Cat)
15	14	exlimiv 1937	. 2 ⊢ (∃𝑥 𝑥 ∈ (Sect‘𝐶) → 𝐶 ∈ Cat)
16	1, 12, 15	3syl 18	1 ⊢ (𝜑 → 𝐶 ∈ Cat)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1547  ∃wex 1786   ∈ wcel 2119   ≠ wne 2935  [wsbc 3730  ∅c0 4268  ⟨cop 4568   class class class wbr 5079  {copab 5141  ‘cfv 6492  (class class class)co 7363   ∈ cmpo 7365  Basecbs 17177  Hom chom 17229  compcco 17230  Catccat 17628  Idccid 17629  Sectcsect 17709

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-sect 17712

This theorem is referenced by:  sectrcl2  49514  isinv2  49517  catcsect  49889