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Theorem subcrcl 17863
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 17859 . 2 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)))})
21mptrcl 7024 1 (𝐻 ∈ (Subcat‘𝐶) → 𝐶 ∈ Cat)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wcel 2105  {cab 2711  wral 3058  [wsbc 3790  cop 4636   class class class wbr 5147  dom cdm 5688  cfv 6562  (class class class)co 7430  compcco 17309  Catccat 17708  Idccid 17709  Homf chomf 17710  cat cssc 17854  Subcatcsubc 17856
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-mpt 5231  df-xp 5694  df-rel 5695  df-cnv 5696  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fv 6570  df-subc 17859
This theorem is referenced by:  subcssc  17890  subcidcl  17894  subccocl  17895  subccatid  17896  subsubc  17903  funcres2b  17947  funcres2  17948  idfusubc  17950
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