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Theorem subcrcl 17078
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 17074 . 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 6770 1 (𝐻 ∈ (Subcat‘𝐶) → 𝐶 ∈ Cat)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 398  wcel 2107  {cab 2797  wral 3136  [wsbc 3770  cop 4565   class class class wbr 5057  dom cdm 5548  cfv 6348  (class class class)co 7148  compcco 16569  Catccat 16927  Idccid 16928  Homf chomf 16929  cat cssc 17069  Subcatcsubc 17071
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-xp 5554  df-rel 5555  df-cnv 5556  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fv 6356  df-subc 17074
This theorem is referenced by:  subcssc  17102  subcidcl  17106  subccocl  17107  subccatid  17108  subsubc  17115  funcres2b  17159  funcres2  17160  idfusubc  44117
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