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Theorem obsubc2 25953
Description: The objects of a subcategory are a subset of those of the supercategory. JFM CAT2 th. 11 . (Contributed by FL, 17-Sep-2009.)
Hypotheses
Ref Expression
obsubc2.1  |-  O1  =  dom  ( id_ `  T
)
obsubc2.2  |-  O 2  =  dom  ( id_ `  U
)
Assertion
Ref Expression
obsubc2  |-  ( U  e.  (  SubCat  `  T
)  ->  O 2  C_  O1 )

Proof of Theorem obsubc2
StepHypRef Expression
1 besubbeca 25951 . 2  |-  ( U  e.  (  SubCat  `  T
)  ->  T  e.  Cat OLD  )
2 eqid 2296 . . . . . 6  |-  ( dom_ `  T )  =  (
dom_ `  T )
3 eqid 2296 . . . . . 6  |-  ( cod_ `  T )  =  (
cod_ `  T )
4 eqid 2296 . . . . . 6  |-  ( o_
`  T )  =  ( o_ `  T
)
5 eqid 2296 . . . . . 6  |-  ( id_ `  T )  =  ( id_ `  T )
62, 3, 4, 5issubcata 25949 . . . . 5  |-  ( T  e.  Cat OLD  ->  ( U  e.  (  SubCat  `  T )  <->  ( U  e.  Cat OLD  /\  (
( id_ `  U
)  C_  ( id_ `  T )  /\  (
( dom_ `  U )  C_  ( dom_ `  T
)  /\  ( cod_ `  U )  C_  ( cod_ `  T ) )  /\  ( o_ `  U )  C_  (
o_ `  T )
) ) ) )
7 dmss 4894 . . . . . . 7  |-  ( ( id_ `  U ) 
C_  ( id_ `  T
)  ->  dom  ( id_ `  U )  C_  dom  ( id_ `  T ) )
873ad2ant1 976 . . . . . 6  |-  ( ( ( id_ `  U
)  C_  ( id_ `  T )  /\  (
( dom_ `  U )  C_  ( dom_ `  T
)  /\  ( cod_ `  U )  C_  ( cod_ `  T ) )  /\  ( o_ `  U )  C_  (
o_ `  T )
)  ->  dom  ( id_ `  U )  C_  dom  ( id_ `  T ) )
98adantl 452 . . . . 5  |-  ( ( U  e.  Cat OLD  /\  ( ( id_ `  U
)  C_  ( id_ `  T )  /\  (
( dom_ `  U )  C_  ( dom_ `  T
)  /\  ( cod_ `  U )  C_  ( cod_ `  T ) )  /\  ( o_ `  U )  C_  (
o_ `  T )
) )  ->  dom  ( id_ `  U ) 
C_  dom  ( id_ `  T ) )
106, 9syl6bi 219 . . . 4  |-  ( T  e.  Cat OLD  ->  ( U  e.  (  SubCat  `  T )  ->  dom  ( id_ `  U ) 
C_  dom  ( id_ `  T ) ) )
1110imp 418 . . 3  |-  ( ( T  e.  Cat OLD  /\  U  e.  (  SubCat  `  T ) )  ->  dom  ( id_ `  U
)  C_  dom  ( id_ `  T ) )
12 obsubc2.2 . . 3  |-  O 2  =  dom  ( id_ `  U
)
13 obsubc2.1 . . 3  |-  O1  =  dom  ( id_ `  T
)
1411, 12, 133sstr4g 3232 . 2  |-  ( ( T  e.  Cat OLD  /\  U  e.  (  SubCat  `  T ) )  ->  O 2  C_  O1 )
151, 14mpancom 650 1  |-  ( U  e.  (  SubCat  `  T
)  ->  O 2  C_  O1 )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696    C_ wss 3165   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:  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  ax-un 4528
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-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fo 5277  df-fv 5279  df-1st 6138  df-2nd 6139  df-dom_ 25820  df-cod_ 25821  df-id_ 25822  df-cmpa 25823  df-catOLD 25856  df-subcat 25947
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