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Theorem subcrcl 14741
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 14737 . . 3  |- Subcat  =  ( c  e.  Cat  |->  { h  |  ( h 
C_cat  ( Hom f  `  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 5346 . 2  |-  dom Subcat  C_  Cat
3 elfvdm 5728 . 2  |-  ( H  e.  (Subcat `  C
)  ->  C  e.  dom Subcat )
42, 3sseldi 3366 1  |-  ( H  e.  (Subcat `  C
)  ->  C  e.  Cat )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    e. wcel 1756   {cab 2429   A.wral 2727   [.wsbc 3198   <.cop 3895   class class class wbr 4304   dom cdm 4852   ` cfv 5430  (class class class)co 6103  compcco 14262   Catccat 14614   Idccid 14615   Hom f chomf 14616    C_cat cssc 14732  Subcatcsubc 14734
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2620  df-ral 2732  df-rex 2733  df-rab 2736  df-v 2986  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-nul 3650  df-if 3804  df-sn 3890  df-pr 3892  df-op 3896  df-uni 4104  df-br 4305  df-opab 4363  df-mpt 4364  df-xp 4858  df-rel 4859  df-cnv 4860  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-iota 5393  df-fv 5438  df-subc 14737
This theorem is referenced by:  subcssc  14762  subcidcl  14766  subccocl  14767  subccatid  14768  subsubc  14775  funcres2b  14819  funcres2  14820
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