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Theorem besubbeca 25848
Description: Lemma to simplify some subcategories related theorems . (Contributed by FL, 17-Sep-2009.)
Assertion
Ref Expression
besubbeca  |-  ( U  e.  (  SubCat  `  T
)  ->  T  e.  Cat OLD  )

Proof of Theorem besubbeca
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 df-subcat 25844 . . 3  |-  SubCat  =  ( x  e.  Cat OLD  |->  (  Cat OLD  i^i  (
( ~P ( dom_ `  x )  X.  ~P ( cod_ `  x )
)  X.  ( ~P ( id_ `  x
)  X.  ~P (
o_ `  x )
) ) ) )
21dmmptss 5169 . 2  |-  dom  SubCat  C_  Cat OLD
3 elfvdm 5554 . 2  |-  ( U  e.  (  SubCat  `  T
)  ->  T  e.  dom  SubCat  )
42, 3sseldi 3178 1  |-  ( U  e.  (  SubCat  `  T
)  ->  T  e.  Cat OLD  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1684    i^i cin 3151   ~Pcpw 3625    X. cxp 4687   dom cdm 4689   ` cfv 5255   dom_cdom_ 25712   cod_ccod_ 25713   id_cid_ 25714   o_co_ 25715    Cat
OLD ccatOLD 25752    SubCat csubcat 25843
This theorem is referenced by:  obsubc2  25850  idsubc  25851  domsubc  25852  codsubc  25853  subctct  25854  morsubc  25855  cmpsubc  25856  idsubidsup  25857  idsubfun  25858
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-subcat 25844
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