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Theorem besubbeca 25951
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 25947 . . 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 5185 . 2  |-  dom  SubCat  C_  Cat OLD
3 elfvdm 5570 . 2  |-  ( U  e.  (  SubCat  `  T
)  ->  T  e.  dom  SubCat  )
42, 3sseldi 3191 1  |-  ( U  e.  (  SubCat  `  T
)  ->  T  e.  Cat OLD  )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1696    i^i cin 3164   ~Pcpw 3638    X. cxp 4703   dom cdm 4705   ` cfv 5271   dom_cdom_ 25815   cod_ccod_ 25816   id_cid_ 25817   o_co_ 25818    Cat
OLD ccatOLD 25855    SubCat csubcat 25946
This theorem is referenced by:  obsubc2  25953  idsubc  25954  domsubc  25955  codsubc  25956  subctct  25957  morsubc  25958  cmpsubc  25959  idsubidsup  25960  idsubfun  25961
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-subcat 25947
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