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Theorem subcrcl 13693
Description: Reverse closure for the subcategory predicate. (Contributed by Mario Carneiro, 6-Jan-2017.)
Assertion
Ref Expression
subcrcl  |-  ( H  e.  (Subcat `  C
)  ->  C  e.  Cat )

Proof of Theorem subcrcl
Dummy variables  f 
c  g  h  s  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-subc 13689 . . 3  |- Subcat  =  ( c  e.  Cat  |->  { h  |  ( h 
C_cat  (  Homf 
`  c )  /\  [.
dom  dom  h  /  s ]. A. x  e.  s  ( ( ( Id
`  c ) `  x )  e.  ( x h x )  /\  A. y  e.  s  A. z  e.  s  A. f  e.  ( x h y ) A. g  e.  ( y h z ) ( g (
<. x ,  y >.
(comp `  c )
z ) f )  e.  ( x h z ) ) ) } )
21dmmptss 5169 . 2  |-  dom Subcat  C_  Cat
3 elfvdm 5554 . 2  |-  ( H  e.  (Subcat `  C
)  ->  C  e.  dom Subcat )
42, 3sseldi 3178 1  |-  ( H  e.  (Subcat `  C
)  ->  C  e.  Cat )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    e. wcel 1684   {cab 2269   A.wral 2543   [.wsbc 2991   <.cop 3643   class class class wbr 4023   dom cdm 4689   ` cfv 5255  (class class class)co 5858  compcco 13220   Catccat 13566   Idccid 13567    Homf chomf 13568    C_cat cssc 13684  Subcatcsubc 13686
This theorem is referenced by:  subcssc  13714  subcidcl  13718  subccocl  13719  subccatid  13720  subsubc  13727  funcres2b  13771  funcres2  13772
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-xp 4695  df-rel 4696  df-cnv 4697  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fv 5263  df-subc 13689
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