Theorem sectrcl 49767

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 5109	. . . . 5 ⊢ (𝐹(𝑋𝑆𝑌)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝑋𝑆𝑌))
3		df-ov 7413	. . . . . 6 ⊢ (𝑋𝑆𝑌) = (𝑆‘⟨𝑋, 𝑌⟩)
4	3	eleq2i 2853	. . . . 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 49594	. . . 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 3020	. . . 4 ⊢ (𝑆 ≠ ∅ ↔ (Sect‘𝐶) ≠ ∅)
10		n0 4306	. . . 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 17803	. . . 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 6999	. . 3 ⊢ (𝑥 ∈ (Sect‘𝐶) → 𝐶 ∈ Cat)
15	14	exlimiv 1958	. 2 ⊢ (∃𝑥 𝑥 ∈ (Sect‘𝐶) → 𝐶 ∈ Cat)
16	1, 12, 15	3syl 19	1 ⊢ (𝜑 → 𝐶 ∈ Cat)

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1568  ∃wex 1807   ∈ wcel 2141   ≠ wne 2956  [wsbc 3743  ∅c0 4285  ⟨cop 4594   class class class wbr 5108  {copab 5172  ‘cfv 6536  (class class class)co 7410   ∈ cmpo 7412  Basecbs 17268  Hom chom 17320  compcco 17321  Catccat 17719  Idccid 17720  Sectcsect 17800

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5256  ax-nul 5268  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2095  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-rab 3415  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-opab 5173  df-mpt 5192  df-xp 5667  df-rel 5668  df-cnv 5669  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fv 6544  df-ov 7413  df-sect 17803

This theorem is referenced by:  sectrcl2  49768  isinv2  49771  catcsect  50143