Theorem sectrcl 49494

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 5087	. . . . 5 ⊢ (𝐹(𝑋𝑆𝑌)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝑋𝑆𝑌))
3		df-ov 7361	. . . . . 6 ⊢ (𝑋𝑆𝑌) = (𝑆‘⟨𝑋, 𝑌⟩)
4	3	eleq2i 2829	. . . . 5 ⊢ (⟨𝐹, 𝐺⟩ ∈ (𝑋𝑆𝑌) ↔ ⟨𝐹, 𝐺⟩ ∈ (𝑆‘⟨𝑋, 𝑌⟩))
5	2, 4	bitri 275	. . . 4 ⊢ (𝐹(𝑋𝑆𝑌)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝑆‘⟨𝑋, 𝑌⟩))
6		elfvne0 49321	. . . 4 ⊢ (⟨𝐹, 𝐺⟩ ∈ (𝑆‘⟨𝑋, 𝑌⟩) → 𝑆 ≠ ∅)
7	5, 6	sylbi 217	. . 3 ⊢ (𝐹(𝑋𝑆𝑌)𝐺 → 𝑆 ≠ ∅)
8		sectrcl.s	. . . . 5 ⊢ 𝑆 = (Sect‘𝐶)
9	8	neeq1i 2997	. . . 4 ⊢ (𝑆 ≠ ∅ ↔ (Sect‘𝐶) ≠ ∅)
10		n0 4294	. . . 4 ⊢ ((Sect‘𝐶) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (Sect‘𝐶))
11	9, 10	bitri 275	. . 3 ⊢ (𝑆 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (Sect‘𝐶))
12	7, 11	sylib 218	. 2 ⊢ (𝐹(𝑋𝑆𝑌)𝐺 → ∃𝑥 𝑥 ∈ (Sect‘𝐶))
13		df-sect 17703	. . . 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 6949	. . 3 ⊢ (𝑥 ∈ (Sect‘𝐶) → 𝐶 ∈ Cat)
15	14	exlimiv 1932	. 2 ⊢ (∃𝑥 𝑥 ∈ (Sect‘𝐶) → 𝐶 ∈ Cat)
16	1, 12, 15	3syl 18	1 ⊢ (𝜑 → 𝐶 ∈ Cat)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542  ∃wex 1781   ∈ wcel 2114   ≠ wne 2933  [wsbc 3729  ∅c0 4274  ⟨cop 4574   class class class wbr 5086  {copab 5148  ‘cfv 6490  (class class class)co 7358   ∈ cmpo 7360  Basecbs 17168  Hom chom 17220  compcco 17221  Catccat 17619  Idccid 17620  Sectcsect 17700

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-nul 5241  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-mpt 5168  df-xp 5628  df-rel 5629  df-cnv 5630  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-iota 6446  df-fv 6498  df-ov 7361  df-sect 17703

This theorem is referenced by:  sectrcl2  49495  isinv2  49498  catcsect  49870