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Theorem catsbc 25952
Description: A category belongs to the set of its subcategories. (Contributed by FL, 17-Sep-2009.)
Assertion
Ref Expression
catsbc  |-  ( T  e.  Cat OLD  ->  T  e.  (  SubCat  `  T
) )

Proof of Theorem catsbc
StepHypRef Expression
1 ssid 3210 . . 3  |-  ( id_ `  T )  C_  ( id_ `  T )
2 ssid 3210 . . . 4  |-  ( dom_ `  T )  C_  ( dom_ `  T )
3 ssid 3210 . . . 4  |-  ( cod_ `  T )  C_  ( cod_ `  T )
42, 3pm3.2i 441 . . 3  |-  ( (
dom_ `  T )  C_  ( dom_ `  T
)  /\  ( cod_ `  T )  C_  ( cod_ `  T ) )
5 ssid 3210 . . 3  |-  ( o_
`  T )  C_  ( o_ `  T )
61, 4, 53pm3.2i 1130 . 2  |-  ( ( id_ `  T ) 
C_  ( id_ `  T
)  /\  ( ( dom_ `  T )  C_  ( dom_ `  T )  /\  ( cod_ `  T
)  C_  ( cod_ `  T ) )  /\  ( o_ `  T ) 
C_  ( o_ `  T ) )
7 eqid 2296 . . . 4  |-  ( dom_ `  T )  =  (
dom_ `  T )
8 eqid 2296 . . . 4  |-  ( cod_ `  T )  =  (
cod_ `  T )
9 eqid 2296 . . . 4  |-  ( o_
`  T )  =  ( o_ `  T
)
10 eqid 2296 . . . 4  |-  ( id_ `  T )  =  ( id_ `  T )
117, 8, 9, 10issubcatb 25950 . . 3  |-  ( ( T  e.  Cat OLD  /\  T  e.  Cat OLD  )  ->  ( T  e.  (  SubCat  `  T )  <->  ( ( id_ `  T
)  C_  ( id_ `  T )  /\  (
( dom_ `  T )  C_  ( dom_ `  T
)  /\  ( cod_ `  T )  C_  ( cod_ `  T ) )  /\  ( o_ `  T )  C_  (
o_ `  T )
) ) )
1211anidms 626 . 2  |-  ( T  e.  Cat OLD  ->  ( T  e.  (  SubCat  `  T )  <->  ( ( id_ `  T )  C_  ( id_ `  T )  /\  ( ( dom_ `  T )  C_  ( dom_ `  T )  /\  ( cod_ `  T )  C_  ( cod_ `  T
) )  /\  (
o_ `  T )  C_  ( o_ `  T
) ) ) )
136, 12mpbiri 224 1  |-  ( T  e.  Cat OLD  ->  T  e.  (  SubCat  `  T
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    e. wcel 1696    C_ wss 3165   ` cfv 5271   dom_cdom_ 25815   cod_ccod_ 25816   id_cid_ 25817   o_co_ 25818    Cat
OLD ccatOLD 25855    SubCat csubcat 25946
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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