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Theorem issubcat 25845
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 5525 . . . . . . 7  |-  ( x  =  T  ->  ( dom_ `  x )  =  ( dom_ `  T
) )
2 issubcat.2 . . . . . . 7  |-  D  =  ( dom_ `  T
)
31, 2syl6eqr 2333 . . . . . 6  |-  ( x  =  T  ->  ( dom_ `  x )  =  D )
43pweqd 3630 . . . . 5  |-  ( x  =  T  ->  ~P ( dom_ `  x )  =  ~P D )
5 fveq2 5525 . . . . . . 7  |-  ( x  =  T  ->  ( cod_ `  x )  =  ( cod_ `  T
) )
6 issubcat.3 . . . . . . 7  |-  C  =  ( cod_ `  T
)
75, 6syl6eqr 2333 . . . . . 6  |-  ( x  =  T  ->  ( cod_ `  x )  =  C )
87pweqd 3630 . . . . 5  |-  ( x  =  T  ->  ~P ( cod_ `  x )  =  ~P C )
94, 8xpeq12d 4714 . . . 4  |-  ( x  =  T  ->  ( ~P ( dom_ `  x
)  X.  ~P ( cod_ `  x ) )  =  ( ~P D  X.  ~P C ) )
10 fveq2 5525 . . . . . . 7  |-  ( x  =  T  ->  ( id_ `  x )  =  ( id_ `  T
) )
11 issubcat.5 . . . . . . 7  |-  J  =  ( id_ `  T
)
1210, 11syl6eqr 2333 . . . . . 6  |-  ( x  =  T  ->  ( id_ `  x )  =  J )
1312pweqd 3630 . . . . 5  |-  ( x  =  T  ->  ~P ( id_ `  x )  =  ~P J )
14 fveq2 5525 . . . . . . 7  |-  ( x  =  T  ->  (
o_ `  x )  =  ( o_ `  T ) )
15 issubcat.4 . . . . . . 7  |-  R  =  ( o_ `  T
)
1614, 15syl6eqr 2333 . . . . . 6  |-  ( x  =  T  ->  (
o_ `  x )  =  R )
1716pweqd 3630 . . . . 5  |-  ( x  =  T  ->  ~P ( o_ `  x )  =  ~P R )
1813, 17xpeq12d 4714 . . . 4  |-  ( x  =  T  ->  ( ~P ( id_ `  x
)  X.  ~P (
o_ `  x )
)  =  ( ~P J  X.  ~P R
) )
199, 18xpeq12d 4714 . . 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 3370 . 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 25844 . 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 5539 . . . . . . 7  |-  ( dom_ `  T )  e.  _V
232, 22eqeltri 2353 . . . . . 6  |-  D  e. 
_V
2423pwex 4193 . . . . 5  |-  ~P D  e.  _V
25 fvex 5539 . . . . . . 7  |-  ( cod_ `  T )  e.  _V
266, 25eqeltri 2353 . . . . . 6  |-  C  e. 
_V
2726pwex 4193 . . . . 5  |-  ~P C  e.  _V
2824, 27xpex 4801 . . . 4  |-  ( ~P D  X.  ~P C
)  e.  _V
29 fvex 5539 . . . . . . 7  |-  ( id_ `  T )  e.  _V
3011, 29eqeltri 2353 . . . . . 6  |-  J  e. 
_V
3130pwex 4193 . . . . 5  |-  ~P J  e.  _V
32 fvex 5539 . . . . . . 7  |-  ( o_
`  T )  e. 
_V
3315, 32eqeltri 2353 . . . . . 6  |-  R  e. 
_V
3433pwex 4193 . . . . 5  |-  ~P R  e.  _V
3531, 34xpex 4801 . . . 4  |-  ( ~P J  X.  ~P R
)  e.  _V
3628, 35xpex 4801 . . 3  |-  ( ( ~P D  X.  ~P C )  X.  ( ~P J  X.  ~P R
) )  e.  _V
3736inex2 4156 . 2  |-  (  Cat
OLD  i^i  ( ( ~P D  X.  ~P C
)  X.  ( ~P J  X.  ~P R
) ) )  e. 
_V
3820, 21, 37fvmpt 5602 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 1623    e. wcel 1684   _Vcvv 2788    i^i cin 3151   ~Pcpw 3625    X. cxp 4687   ` 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:  issubcata  25846
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  ax-un 4512
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-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-subcat 25844
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