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Theorem invrcl 49654
Description: Reverse closure for inverse relations. (Contributed by Zhi Wang, 14-Nov-2025.)
Hypotheses
Ref Expression
invrcl.n 𝑁 = (Inv‘𝐶)
invrcl.f (𝜑𝐹(𝑋𝑁𝑌)𝐺)
Assertion
Ref Expression
invrcl (𝜑𝐶 ∈ Cat)

Proof of Theorem invrcl
Dummy variables 𝑥 𝑦 𝑐 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 invrcl.f . 2 (𝜑𝐹(𝑋𝑁𝑌)𝐺)
2 df-br 5105 . . . . 5 (𝐹(𝑋𝑁𝑌)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝑋𝑁𝑌))
3 df-ov 7403 . . . . . 6 (𝑋𝑁𝑌) = (𝑁‘⟨𝑋, 𝑌⟩)
43eleq2i 2857 . . . . 5 (⟨𝐹, 𝐺⟩ ∈ (𝑋𝑁𝑌) ↔ ⟨𝐹, 𝐺⟩ ∈ (𝑁‘⟨𝑋, 𝑌⟩))
52, 4bitri 278 . . . 4 (𝐹(𝑋𝑁𝑌)𝐺 ↔ ⟨𝐹, 𝐺⟩ ∈ (𝑁‘⟨𝑋, 𝑌⟩))
6 elfvne0 49479 . . . 4 (⟨𝐹, 𝐺⟩ ∈ (𝑁‘⟨𝑋, 𝑌⟩) → 𝑁 ≠ ∅)
75, 6sylbi 220 . . 3 (𝐹(𝑋𝑁𝑌)𝐺𝑁 ≠ ∅)
8 invrcl.n . . . . 5 𝑁 = (Inv‘𝐶)
98neeq1i 3024 . . . 4 (𝑁 ≠ ∅ ↔ (Inv‘𝐶) ≠ ∅)
10 n0 4308 . . . 4 ((Inv‘𝐶) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (Inv‘𝐶))
119, 10bitri 278 . . 3 (𝑁 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ (Inv‘𝐶))
127, 11sylib 221 . 2 (𝐹(𝑋𝑁𝑌)𝐺 → ∃𝑥 𝑥 ∈ (Inv‘𝐶))
13 df-inv 17793 . . . 4 Inv = (𝑐 ∈ Cat ↦ (𝑥 ∈ (Base‘𝑐), 𝑦 ∈ (Base‘𝑐) ↦ ((𝑥(Sect‘𝑐)𝑦) ∩ (𝑦(Sect‘𝑐)𝑥))))
1413mptrcl 6989 . . 3 (𝑥 ∈ (Inv‘𝐶) → 𝐶 ∈ Cat)
1514exlimiv 1953 . 2 (∃𝑥 𝑥 ∈ (Inv‘𝐶) → 𝐶 ∈ Cat)
161, 12, 153syl 19 1 (𝜑𝐶 ∈ Cat)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1563  wex 1802  wcel 2145  wne 2960  cin 3906  c0 4288  cop 4591   class class class wbr 5104  ccnv 5650  cfv 6525  (class class class)co 7400  cmpo 7402  Basecbs 17257  Catccat 17708  Sectcsect 17789  Invcinv 17790
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 5250  ax-nul 5260  ax-pr 5394
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 4868  df-br 5105  df-opab 5167  df-mpt 5186  df-xp 5657  df-rel 5658  df-cnv 5659  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6481  df-fv 6533  df-ov 7403  df-inv 17793
This theorem is referenced by:  invrcl2  49655  isinv2  49656  isoval2  49665
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