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Theorem subcrcl 13709
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 13705 . . 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 5185 . 2  |-  dom Subcat  C_  Cat
3 elfvdm 5570 . 2  |-  ( H  e.  (Subcat `  C
)  ->  C  e.  dom Subcat )
42, 3sseldi 3191 1  |-  ( H  e.  (Subcat `  C
)  ->  C  e.  Cat )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    e. wcel 1696   {cab 2282   A.wral 2556   [.wsbc 3004   <.cop 3656   class class class wbr 4039   dom cdm 4705   ` cfv 5271  (class class class)co 5874  compcco 13236   Catccat 13582   Idccid 13583    Homf chomf 13584    C_cat cssc 13700  Subcatcsubc 13702
This theorem is referenced by:  subcssc  13730  subcidcl  13734  subccocl  13735  subccatid  13736  subsubc  13743  funcres2b  13787  funcres2  13788
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-xp 4711  df-rel 4712  df-cnv 4713  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fv 5279  df-subc 13705
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