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Theorem sscpwex 13708
Description: An analogue of pwex 4209 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 5899 . 2  |-  ( ~P
U. ran  J  ^pm  dom 
J )  e.  _V
2 brssc 13707 . . . 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 443 . . . . . . . . . . 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 2804 . . . . . . . . . . . 12  |-  t  e. 
_V
54, 4xpex 4817 . . . . . . . . . . 11  |-  ( t  X.  t )  e. 
_V
6 fnex 5757 . . . . . . . . . . 11  |-  ( ( J  Fn  ( t  X.  t )  /\  ( t  X.  t
)  e.  _V )  ->  J  e.  _V )
73, 5, 6sylancl 643 . . . . . . . . . 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 4956 . . . . . . . . . 10  |-  ( J  e.  _V  ->  ran  J  e.  _V )
97, 8syl 15 . . . . . . . . 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 4533 . . . . . . . . 9  |-  ( ran 
J  e.  _V  ->  U.
ran  J  e.  _V )
11 pwexg 4210 . . . . . . . . 9  |-  ( U. ran  J  e.  _V  ->  ~P
U. ran  J  e.  _V )
129, 10, 113syl 18 . . . . . . . 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 5359 . . . . . . . . . 10  |-  ( J  Fn  ( t  X.  t )  ->  dom  J  =  ( t  X.  t ) )
1413adantr 451 . . . . . . . . 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 2384 . . . . . . . 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 6845 . . . . . . . . . . 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 5567 . . . . . . . . . . . . 13  |-  ( J `
 x )  C_  U.
ran  J
18 sspwb 4239 . . . . . . . . . . . . 13  |-  ( ( J `  x ) 
C_  U. ran  J  <->  ~P ( J `  x )  C_ 
~P U. ran  J )
1917, 18mpbi 199 . . . . . . . . . . . 12  |-  ~P ( J `  x )  C_ 
~P U. ran  J
2019a1i 10 . . . . . . . . . . 11  |-  ( x  e.  ( s  X.  s )  ->  ~P ( J `  x ) 
C_  ~P U. ran  J
)
2116, 20mprg 2625 . . . . . . . . . 10  |-  X_ x  e.  ( s  X.  s
) ~P ( J `
 x )  C_  X_ x  e.  ( s  X.  s ) ~P
U. ran  J
22 simprr 733 . . . . . . . . . 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 3191 . . . . . . . . 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 2804 . . . . . . . . . 10  |-  h  e. 
_V
2524elixpconst 6840 . . . . . . . . 9  |-  ( h  e.  X_ x  e.  ( s  X.  s ) ~P U. ran  J  <->  h : ( s  X.  s ) --> ~P U. ran  J )
2623, 25sylib 188 . . . . . . . 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 3646 . . . . . . . . . . 11  |-  ( s  e.  ~P t  -> 
s  C_  t )
2827ad2antrl 708 . . . . . . . . . 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 4808 . . . . . . . . . 10  |-  ( ( s  C_  t  /\  s  C_  t )  -> 
( s  X.  s
)  C_  ( t  X.  t ) )
3028, 28, 29syl2anc 642 . . . . . . . . 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 3228 . . . . . . . 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 6804 . . . . . . . 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 1183 . . . . . . 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 2681 . . . . . 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 418 . . . . 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 1624 . . . 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 187 . . 3  |-  ( h 
C_cat  J  ->  h  e.  ( ~P U. ran  J  ^pm  dom  J ) )
3837abssi 3261 . 2  |-  { h  |  h  C_cat  J }  C_  ( ~P U. ran  J 
^pm  dom  J )
391, 38ssexi 4175 1  |-  { h  |  h  C_cat  J }  e.  _V
Colors of variables: wff set class
Syntax hints:    /\ wa 358   E.wex 1531    = wceq 1632    e. wcel 1696   {cab 2282   E.wrex 2557   _Vcvv 2801    C_ wss 3165   ~Pcpw 3638   U.cuni 3843   class class class wbr 4039    X. cxp 4703   dom cdm 4705   ran crn 4706    Fn wfn 5266   -->wf 5267   ` cfv 5271  (class class class)co 5874    ^pm cpm 6789   X_cixp 6833    C_cat cssc 13700
This theorem is referenced by:  issubc  13728
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-rep 4147  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-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  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-iun 3923  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-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-pm 6791  df-ixp 6834  df-ssc 13703
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