Theorem sectrcl 49651

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 5106	. . . . 5 ⊢ (𝐹(𝑋𝑆𝑌)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝑋𝑆𝑌))
3		df-ov 7403	. . . . . 6 ⊢ (𝑋𝑆𝑌) = (𝑆‘⟨𝑋, 𝑌⟩)
4	3	eleq2i 2857	. . . . 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 49478	. . . 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 3024	. . . 4 ⊢ (𝑆 ≠ ∅ ↔ (Sect‘𝐶) ≠ ∅)
10		n0 4308	. . . 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 17794	. . . 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 6989	. . 3 ⊢ (𝑥 ∈ (Sect‘𝐶) → 𝐶 ∈ Cat)
15	14	exlimiv 1953	. 2 ⊢ (∃𝑥 𝑥 ∈ (Sect‘𝐶) → 𝐶 ∈ Cat)
16	1, 12, 15	3syl 19	1 ⊢ (𝜑 → 𝐶 ∈ Cat)

Syntax hints:   → wi 4   ∧ wa 400   = wceq 1563  ∃wex 1802   ∈ wcel 2145   ≠ wne 2960  [wsbc 3747  ∅c0 4288  ⟨cop 4591   class class class wbr 5105  {copab 5167  ‘cfv 6525  (class class class)co 7400   ∈ cmpo 7402  Basecbs 17259  Hom chom 17311  compcco 17312  Catccat 17710  Idccid 17711  Sectcsect 17791

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5251  ax-nul 5261  ax-pr 5395

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4869  df-br 5106  df-opab 5168  df-mpt 5187  df-xp 5658  df-rel 5659  df-cnv 5660  df-dm 5662  df-rn 5663  df-res 5664  df-ima 5665  df-iota 6481  df-fv 6533  df-ov 7403  df-sect 17794

This theorem is referenced by:  sectrcl2  49652  isinv2  49655  catcsect  50027