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Theorem sscpwex 14015
Description: An analogue of pwex 4382 for the subcategory subset relation: The collection of subcategory subsets of a given set  J is a set. (Contributed by Mario Carneiro, 6-Jan-2017.)
Assertion
Ref Expression
sscpwex  |-  { h  |  h  C_cat  J }  e.  _V
Distinct variable group:    h, J

Proof of Theorem sscpwex
Dummy variables  s 
t  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ovex 6106 . 2  |-  ( ~P
U. ran  J  ^pm  dom 
J )  e.  _V
2 brssc 14014 . . . 4  |-  ( h 
C_cat  J  <->  E. t ( J  Fn  ( t  X.  t )  /\  E. s  e.  ~P  t
h  e.  X_ x  e.  ( s  X.  s
) ~P ( J `
 x ) ) )
3 simpl 444 . . . . . . . . . . 11  |-  ( ( J  Fn  ( t  X.  t )  /\  ( s  e.  ~P t  /\  h  e.  X_ x  e.  ( s  X.  s ) ~P ( J `  x )
) )  ->  J  Fn  ( t  X.  t
) )
4 vex 2959 . . . . . . . . . . . 12  |-  t  e. 
_V
54, 4xpex 4990 . . . . . . . . . . 11  |-  ( t  X.  t )  e. 
_V
6 fnex 5961 . . . . . . . . . . 11  |-  ( ( J  Fn  ( t  X.  t )  /\  ( t  X.  t
)  e.  _V )  ->  J  e.  _V )
73, 5, 6sylancl 644 . . . . . . . . . 10  |-  ( ( J  Fn  ( t  X.  t )  /\  ( s  e.  ~P t  /\  h  e.  X_ x  e.  ( s  X.  s ) ~P ( J `  x )
) )  ->  J  e.  _V )
8 rnexg 5131 . . . . . . . . . 10  |-  ( J  e.  _V  ->  ran  J  e.  _V )
97, 8syl 16 . . . . . . . . 9  |-  ( ( J  Fn  ( t  X.  t )  /\  ( s  e.  ~P t  /\  h  e.  X_ x  e.  ( s  X.  s ) ~P ( J `  x )
) )  ->  ran  J  e.  _V )
10 uniexg 4706 . . . . . . . . 9  |-  ( ran 
J  e.  _V  ->  U.
ran  J  e.  _V )
11 pwexg 4383 . . . . . . . . 9  |-  ( U. ran  J  e.  _V  ->  ~P
U. ran  J  e.  _V )
129, 10, 113syl 19 . . . . . . . 8  |-  ( ( J  Fn  ( t  X.  t )  /\  ( s  e.  ~P t  /\  h  e.  X_ x  e.  ( s  X.  s ) ~P ( J `  x )
) )  ->  ~P U.
ran  J  e.  _V )
13 fndm 5544 . . . . . . . . . 10  |-  ( J  Fn  ( t  X.  t )  ->  dom  J  =  ( t  X.  t ) )
1413adantr 452 . . . . . . . . 9  |-  ( ( J  Fn  ( t  X.  t )  /\  ( s  e.  ~P t  /\  h  e.  X_ x  e.  ( s  X.  s ) ~P ( J `  x )
) )  ->  dom  J  =  ( t  X.  t ) )
1514, 5syl6eqel 2524 . . . . . . . 8  |-  ( ( J  Fn  ( t  X.  t )  /\  ( s  e.  ~P t  /\  h  e.  X_ x  e.  ( s  X.  s ) ~P ( J `  x )
) )  ->  dom  J  e.  _V )
16 ss2ixp 7075 . . . . . . . . . . 11  |-  ( A. x  e.  ( s  X.  s ) ~P ( J `  x )  C_ 
~P U. ran  J  ->  X_ x  e.  ( s  X.  s ) ~P ( J `  x
)  C_  X_ x  e.  ( s  X.  s
) ~P U. ran  J )
17 fvssunirn 5754 . . . . . . . . . . . . 13  |-  ( J `
 x )  C_  U.
ran  J
18 sspwb 4413 . . . . . . . . . . . . 13  |-  ( ( J `  x ) 
C_  U. ran  J  <->  ~P ( J `  x )  C_ 
~P U. ran  J )
1917, 18mpbi 200 . . . . . . . . . . . 12  |-  ~P ( J `  x )  C_ 
~P U. ran  J
2019a1i 11 . . . . . . . . . . 11  |-  ( x  e.  ( s  X.  s )  ->  ~P ( J `  x ) 
C_  ~P U. ran  J
)
2116, 20mprg 2775 . . . . . . . . . 10  |-  X_ x  e.  ( s  X.  s
) ~P ( J `
 x )  C_  X_ x  e.  ( s  X.  s ) ~P
U. ran  J
22 simprr 734 . . . . . . . . . 10  |-  ( ( J  Fn  ( t  X.  t )  /\  ( s  e.  ~P t  /\  h  e.  X_ x  e.  ( s  X.  s ) ~P ( J `  x )
) )  ->  h  e.  X_ x  e.  ( s  X.  s ) ~P ( J `  x ) )
2321, 22sseldi 3346 . . . . . . . . 9  |-  ( ( J  Fn  ( t  X.  t )  /\  ( s  e.  ~P t  /\  h  e.  X_ x  e.  ( s  X.  s ) ~P ( J `  x )
) )  ->  h  e.  X_ x  e.  ( s  X.  s ) ~P U. ran  J
)
24 vex 2959 . . . . . . . . . 10  |-  h  e. 
_V
2524elixpconst 7070 . . . . . . . . 9  |-  ( h  e.  X_ x  e.  ( s  X.  s ) ~P U. ran  J  <->  h : ( s  X.  s ) --> ~P U. ran  J )
2623, 25sylib 189 . . . . . . . 8  |-  ( ( J  Fn  ( t  X.  t )  /\  ( s  e.  ~P t  /\  h  e.  X_ x  e.  ( s  X.  s ) ~P ( J `  x )
) )  ->  h : ( s  X.  s ) --> ~P U. ran  J )
27 elpwi 3807 . . . . . . . . . . 11  |-  ( s  e.  ~P t  -> 
s  C_  t )
2827ad2antrl 709 . . . . . . . . . 10  |-  ( ( J  Fn  ( t  X.  t )  /\  ( s  e.  ~P t  /\  h  e.  X_ x  e.  ( s  X.  s ) ~P ( J `  x )
) )  ->  s  C_  t )
29 xpss12 4981 . . . . . . . . . 10  |-  ( ( s  C_  t  /\  s  C_  t )  -> 
( s  X.  s
)  C_  ( t  X.  t ) )
3028, 28, 29syl2anc 643 . . . . . . . . 9  |-  ( ( J  Fn  ( t  X.  t )  /\  ( s  e.  ~P t  /\  h  e.  X_ x  e.  ( s  X.  s ) ~P ( J `  x )
) )  ->  (
s  X.  s ) 
C_  ( t  X.  t ) )
3130, 14sseqtr4d 3385 . . . . . . . 8  |-  ( ( J  Fn  ( t  X.  t )  /\  ( s  e.  ~P t  /\  h  e.  X_ x  e.  ( s  X.  s ) ~P ( J `  x )
) )  ->  (
s  X.  s ) 
C_  dom  J )
32 elpm2r 7034 . . . . . . . 8  |-  ( ( ( ~P U. ran  J  e.  _V  /\  dom  J  e.  _V )  /\  ( h : ( s  X.  s ) --> ~P U. ran  J  /\  ( s  X.  s
)  C_  dom  J ) )  ->  h  e.  ( ~P U. ran  J  ^pm  dom  J ) )
3312, 15, 26, 31, 32syl22anc 1185 . . . . . . 7  |-  ( ( J  Fn  ( t  X.  t )  /\  ( s  e.  ~P t  /\  h  e.  X_ x  e.  ( s  X.  s ) ~P ( J `  x )
) )  ->  h  e.  ( ~P U. ran  J 
^pm  dom  J ) )
3433rexlimdvaa 2831 . . . . . 6  |-  ( J  Fn  ( t  X.  t )  ->  ( E. s  e.  ~P  t h  e.  X_ x  e.  ( s  X.  s
) ~P ( J `
 x )  ->  h  e.  ( ~P U.
ran  J  ^pm  dom  J
) ) )
3534imp 419 . . . . 5  |-  ( ( J  Fn  ( t  X.  t )  /\  E. s  e.  ~P  t
h  e.  X_ x  e.  ( s  X.  s
) ~P ( J `
 x ) )  ->  h  e.  ( ~P U. ran  J  ^pm  dom  J ) )
3635exlimiv 1644 . . . 4  |-  ( E. t ( J  Fn  ( t  X.  t
)  /\  E. s  e.  ~P  t h  e.  X_ x  e.  (
s  X.  s ) ~P ( J `  x ) )  ->  h  e.  ( ~P U.
ran  J  ^pm  dom  J
) )
372, 36sylbi 188 . . 3  |-  ( h 
C_cat  J  ->  h  e.  ( ~P U. ran  J  ^pm  dom  J ) )
3837abssi 3418 . 2  |-  { h  |  h  C_cat  J }  C_  ( ~P U. ran  J 
^pm  dom  J )
391, 38ssexi 4348 1  |-  { h  |  h  C_cat  J }  e.  _V
Colors of variables: wff set class
Syntax hints:    /\ wa 359   E.wex 1550    = wceq 1652    e. wcel 1725   {cab 2422   E.wrex 2706   _Vcvv 2956    C_ wss 3320   ~Pcpw 3799   U.cuni 4015   class class class wbr 4212    X. cxp 4876   dom cdm 4878   ran crn 4879    Fn wfn 5449   -->wf 5450   ` cfv 5454  (class class class)co 6081    ^pm cpm 7019   X_cixp 7063    C_cat cssc 14007
This theorem is referenced by:  issubc  14035
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-pm 7021  df-ixp 7064  df-ssc 14010
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