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Theorem sscpwex 14724
Description: An analogue of pwex 4472 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 6115 . 2  |-  ( ~P
U. ran  J  ^pm  dom 
J )  e.  _V
2 brssc 14723 . . . 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 454 . . . . . . . . . 10  |-  ( ( 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 2973 . . . . . . . . . . 11  |-  t  e. 
_V
54, 4xpex 6507 . . . . . . . . . 10  |-  ( t  X.  t )  e. 
_V
6 fnex 5941 . . . . . . . . . 10  |-  ( ( J  Fn  ( t  X.  t )  /\  ( t  X.  t
)  e.  _V )  ->  J  e.  _V )
73, 5, 6sylancl 657 . . . . . . . . 9  |-  ( ( 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 6509 . . . . . . . . 9  |-  ( J  e.  _V  ->  ran  J  e.  _V )
9 uniexg 6376 . . . . . . . . 9  |-  ( ran 
J  e.  _V  ->  U.
ran  J  e.  _V )
10 pwexg 4473 . . . . . . . . 9  |-  ( U. ran  J  e.  _V  ->  ~P
U. ran  J  e.  _V )
117, 8, 9, 104syl 21 . . . . . . . 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 )
12 fndm 5507 . . . . . . . . . 10  |-  ( J  Fn  ( t  X.  t )  ->  dom  J  =  ( t  X.  t ) )
1312adantr 462 . . . . . . . . 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 ) )
1413, 5syl6eqel 2529 . . . . . . . 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 )
15 ss2ixp 7272 . . . . . . . . . . 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 )
16 fvssunirn 5710 . . . . . . . . . . . . 13  |-  ( J `
 x )  C_  U.
ran  J
17 sspwb 4538 . . . . . . . . . . . . 13  |-  ( ( J `  x ) 
C_  U. ran  J  <->  ~P ( J `  x )  C_ 
~P U. ran  J )
1816, 17mpbi 208 . . . . . . . . . . . 12  |-  ~P ( J `  x )  C_ 
~P U. ran  J
1918a1i 11 . . . . . . . . . . 11  |-  ( x  e.  ( s  X.  s )  ->  ~P ( J `  x ) 
C_  ~P U. ran  J
)
2015, 19mprg 2783 . . . . . . . . . 10  |-  X_ x  e.  ( s  X.  s
) ~P ( J `
 x )  C_  X_ x  e.  ( s  X.  s ) ~P
U. ran  J
21 simprr 751 . . . . . . . . . 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 ) )
2220, 21sseldi 3351 . . . . . . . . 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
)
23 vex 2973 . . . . . . . . . 10  |-  h  e. 
_V
2423elixpconst 7267 . . . . . . . . 9  |-  ( h  e.  X_ x  e.  ( s  X.  s ) ~P U. ran  J  <->  h : ( s  X.  s ) --> ~P U. ran  J )
2522, 24sylib 196 . . . . . . . 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 )
26 elpwi 3866 . . . . . . . . . . 11  |-  ( s  e.  ~P t  -> 
s  C_  t )
2726ad2antrl 722 . . . . . . . . . 10  |-  ( ( J  Fn  ( t  X.  t )  /\  ( s  e.  ~P t  /\  h  e.  X_ x  e.  ( s  X.  s ) ~P ( J `  x )
) )  ->  s  C_  t )
28 xpss12 4941 . . . . . . . . . 10  |-  ( ( s  C_  t  /\  s  C_  t )  -> 
( s  X.  s
)  C_  ( t  X.  t ) )
2927, 27, 28syl2anc 656 . . . . . . . . 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 ) )
3029, 13sseqtr4d 3390 . . . . . . . 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 )
31 elpm2r 7226 . . . . . . . 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 ) )
3211, 14, 25, 30, 31syl22anc 1214 . . . . . . 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 ) )
3332rexlimdvaa 2840 . . . . . 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
) ) )
3433imp 429 . . . . 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 ) )
3534exlimiv 1693 . . . 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
) )
362, 35sylbi 195 . . 3  |-  ( h 
C_cat  J  ->  h  e.  ( ~P U. ran  J  ^pm  dom  J ) )
3736abssi 3424 . 2  |-  { h  |  h  C_cat  J }  C_  ( ~P U. ran  J 
^pm  dom  J )
381, 37ssexi 4434 1  |-  { h  |  h  C_cat  J }  e.  _V
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    = wceq 1364   E.wex 1591    e. wcel 1761   {cab 2427   E.wrex 2714   _Vcvv 2970    C_ wss 3325   ~Pcpw 3857   U.cuni 4088   class class class wbr 4289    X. cxp 4834   dom cdm 4836   ran crn 4837    Fn wfn 5410   -->wf 5411   ` cfv 5415  (class class class)co 6090    ^pm cpm 7211   X_cixp 7259    C_cat cssc 14716
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1713  ax-7 1733  ax-8 1763  ax-9 1765  ax-10 1780  ax-11 1785  ax-12 1797  ax-13 1948  ax-ext 2422  ax-rep 4400  ax-sep 4410  ax-nul 4418  ax-pow 4467  ax-pr 4528  ax-un 6371
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 962  df-tru 1367  df-ex 1592  df-nf 1595  df-sb 1706  df-eu 2263  df-mo 2264  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-reu 2720  df-rab 2722  df-v 2972  df-sbc 3184  df-csb 3286  df-dif 3328  df-un 3330  df-in 3332  df-ss 3339  df-nul 3635  df-if 3789  df-pw 3859  df-sn 3875  df-pr 3877  df-op 3881  df-uni 4089  df-iun 4170  df-br 4290  df-opab 4348  df-mpt 4349  df-id 4632  df-xp 4842  df-rel 4843  df-cnv 4844  df-co 4845  df-dm 4846  df-rn 4847  df-res 4848  df-ima 4849  df-iota 5378  df-fun 5417  df-fn 5418  df-f 5419  df-f1 5420  df-fo 5421  df-f1o 5422  df-fv 5423  df-ov 6093  df-oprab 6094  df-mpt2 6095  df-pm 7213  df-ixp 7260  df-ssc 14719
This theorem is referenced by:  issubc  14744
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