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Theorem subcrcl 16392
Description: Reverse closure for the subcategory predicate. (Contributed by Mario Carneiro, 6-Jan-2017.)
Assertion
Ref Expression
subcrcl (𝐻 ∈ (Subcat‘𝐶) → 𝐶 ∈ Cat)

Proof of Theorem subcrcl
Dummy variables 𝑓 𝑐 𝑔 𝑠 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-subc 16388 . . 3 Subcat = (𝑐 ∈ Cat ↦ { ∣ (cat (Homf𝑐) ∧ [dom dom / 𝑠]𝑥𝑠 (((Id‘𝑐)‘𝑥) ∈ (𝑥𝑥) ∧ ∀𝑦𝑠𝑧𝑠𝑓 ∈ (𝑥𝑦)∀𝑔 ∈ (𝑦𝑧)(𝑔(⟨𝑥, 𝑦⟩(comp‘𝑐)𝑧)𝑓) ∈ (𝑥𝑧)))})
21dmmptss 5593 . 2 dom Subcat ⊆ Cat
3 elfvdm 6178 . 2 (𝐻 ∈ (Subcat‘𝐶) → 𝐶 ∈ dom Subcat)
42, 3sseldi 3586 1 (𝐻 ∈ (Subcat‘𝐶) → 𝐶 ∈ Cat)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wcel 1992  {cab 2612  wral 2912  [wsbc 3422  cop 4159   class class class wbr 4618  dom cdm 5079  cfv 5850  (class class class)co 6605  compcco 15869  Catccat 16241  Idccid 16242  Homf chomf 16243  cat cssc 16383  Subcatcsubc 16385
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-xp 5085  df-rel 5086  df-cnv 5087  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5813  df-fv 5858  df-subc 16388
This theorem is referenced by:  subcssc  16416  subcidcl  16420  subccocl  16421  subccatid  16422  subsubc  16429  funcres2b  16473  funcres2  16474  idfusubc  41127
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