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Theorem issubcat 25948
Description: The set of all the subcategories of  T. (Contributed by FL, 17-Sep-2009.) (Revised by Mario Carneiro, 27-Dec-2014.)
Hypotheses
Ref Expression
issubcat.2  |-  D  =  ( dom_ `  T
)
issubcat.3  |-  C  =  ( cod_ `  T
)
issubcat.4  |-  R  =  ( o_ `  T
)
issubcat.5  |-  J  =  ( id_ `  T
)
Assertion
Ref Expression
issubcat  |-  ( T  e.  Cat OLD  ->  ( 
SubCat  `  T )  =  (  Cat OLD  i^i  ( ( ~P D  X.  ~P C )  X.  ( ~P J  X.  ~P R ) ) ) )

Proof of Theorem issubcat
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 fveq2 5541 . . . . . . 7  |-  ( x  =  T  ->  ( dom_ `  x )  =  ( dom_ `  T
) )
2 issubcat.2 . . . . . . 7  |-  D  =  ( dom_ `  T
)
31, 2syl6eqr 2346 . . . . . 6  |-  ( x  =  T  ->  ( dom_ `  x )  =  D )
43pweqd 3643 . . . . 5  |-  ( x  =  T  ->  ~P ( dom_ `  x )  =  ~P D )
5 fveq2 5541 . . . . . . 7  |-  ( x  =  T  ->  ( cod_ `  x )  =  ( cod_ `  T
) )
6 issubcat.3 . . . . . . 7  |-  C  =  ( cod_ `  T
)
75, 6syl6eqr 2346 . . . . . 6  |-  ( x  =  T  ->  ( cod_ `  x )  =  C )
87pweqd 3643 . . . . 5  |-  ( x  =  T  ->  ~P ( cod_ `  x )  =  ~P C )
94, 8xpeq12d 4730 . . . 4  |-  ( x  =  T  ->  ( ~P ( dom_ `  x
)  X.  ~P ( cod_ `  x ) )  =  ( ~P D  X.  ~P C ) )
10 fveq2 5541 . . . . . . 7  |-  ( x  =  T  ->  ( id_ `  x )  =  ( id_ `  T
) )
11 issubcat.5 . . . . . . 7  |-  J  =  ( id_ `  T
)
1210, 11syl6eqr 2346 . . . . . 6  |-  ( x  =  T  ->  ( id_ `  x )  =  J )
1312pweqd 3643 . . . . 5  |-  ( x  =  T  ->  ~P ( id_ `  x )  =  ~P J )
14 fveq2 5541 . . . . . . 7  |-  ( x  =  T  ->  (
o_ `  x )  =  ( o_ `  T ) )
15 issubcat.4 . . . . . . 7  |-  R  =  ( o_ `  T
)
1614, 15syl6eqr 2346 . . . . . 6  |-  ( x  =  T  ->  (
o_ `  x )  =  R )
1716pweqd 3643 . . . . 5  |-  ( x  =  T  ->  ~P ( o_ `  x )  =  ~P R )
1813, 17xpeq12d 4730 . . . 4  |-  ( x  =  T  ->  ( ~P ( id_ `  x
)  X.  ~P (
o_ `  x )
)  =  ( ~P J  X.  ~P R
) )
199, 18xpeq12d 4730 . . 3  |-  ( x  =  T  ->  (
( ~P ( dom_ `  x )  X.  ~P ( cod_ `  x )
)  X.  ( ~P ( id_ `  x
)  X.  ~P (
o_ `  x )
) )  =  ( ( ~P D  X.  ~P C )  X.  ( ~P J  X.  ~P R
) ) )
2019ineq2d 3383 . 2  |-  ( x  =  T  ->  (  Cat OLD  i^i  ( ( ~P ( dom_ `  x
)  X.  ~P ( cod_ `  x ) )  X.  ( ~P ( id_ `  x )  X. 
~P ( o_ `  x ) ) ) )  =  (  Cat
OLD  i^i  ( ( ~P D  X.  ~P C
)  X.  ( ~P J  X.  ~P R
) ) ) )
21 df-subcat 25947 . 2  |-  SubCat  =  ( x  e.  Cat OLD  |->  (  Cat OLD  i^i  (
( ~P ( dom_ `  x )  X.  ~P ( cod_ `  x )
)  X.  ( ~P ( id_ `  x
)  X.  ~P (
o_ `  x )
) ) ) )
22 fvex 5555 . . . . . . 7  |-  ( dom_ `  T )  e.  _V
232, 22eqeltri 2366 . . . . . 6  |-  D  e. 
_V
2423pwex 4209 . . . . 5  |-  ~P D  e.  _V
25 fvex 5555 . . . . . . 7  |-  ( cod_ `  T )  e.  _V
266, 25eqeltri 2366 . . . . . 6  |-  C  e. 
_V
2726pwex 4209 . . . . 5  |-  ~P C  e.  _V
2824, 27xpex 4817 . . . 4  |-  ( ~P D  X.  ~P C
)  e.  _V
29 fvex 5555 . . . . . . 7  |-  ( id_ `  T )  e.  _V
3011, 29eqeltri 2366 . . . . . 6  |-  J  e. 
_V
3130pwex 4209 . . . . 5  |-  ~P J  e.  _V
32 fvex 5555 . . . . . . 7  |-  ( o_
`  T )  e. 
_V
3315, 32eqeltri 2366 . . . . . 6  |-  R  e. 
_V
3433pwex 4209 . . . . 5  |-  ~P R  e.  _V
3531, 34xpex 4817 . . . 4  |-  ( ~P J  X.  ~P R
)  e.  _V
3628, 35xpex 4817 . . 3  |-  ( ( ~P D  X.  ~P C )  X.  ( ~P J  X.  ~P R
) )  e.  _V
3736inex2 4172 . 2  |-  (  Cat
OLD  i^i  ( ( ~P D  X.  ~P C
)  X.  ( ~P J  X.  ~P R
) ) )  e. 
_V
3820, 21, 37fvmpt 5618 1  |-  ( T  e.  Cat OLD  ->  ( 
SubCat  `  T )  =  (  Cat OLD  i^i  ( ( ~P D  X.  ~P C )  X.  ( ~P J  X.  ~P R ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696   _Vcvv 2801    i^i cin 3164   ~Pcpw 3638    X. cxp 4703   ` 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:  issubcata  25949
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-iota 5235  df-fun 5273  df-fv 5279  df-subcat 25947
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