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Theorem subcrcl 14016
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 14012 . . 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 5366 . 2  |-  dom Subcat  C_  Cat
3 elfvdm 5757 . 2  |-  ( H  e.  (Subcat `  C
)  ->  C  e.  dom Subcat )
42, 3sseldi 3346 1  |-  ( H  e.  (Subcat `  C
)  ->  C  e.  Cat )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    e. wcel 1725   {cab 2422   A.wral 2705   [.wsbc 3161   <.cop 3817   class class class wbr 4212   dom cdm 4878   ` cfv 5454  (class class class)co 6081  compcco 13541   Catccat 13889   Idccid 13890    Homf chomf 13891    C_cat cssc 14007  Subcatcsubc 14009
This theorem is referenced by:  subcssc  14037  subcidcl  14041  subccocl  14042  subccatid  14043  subsubc  14050  funcres2b  14094  funcres2  14095
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-xp 4884  df-rel 4885  df-cnv 4886  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fv 5462  df-subc 14012
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