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Theorem sscpwex 15720
Description: An analogue of pwex 4586 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 6318 . 2  |-  ( ~P
U. ran  J  ^pm  dom 
J )  e.  _V
2 brssc 15719 . . . 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 459 . . . . . . . . . 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 3048 . . . . . . . . . . 11  |-  t  e. 
_V
54, 4xpex 6595 . . . . . . . . . 10  |-  ( t  X.  t )  e. 
_V
6 fnex 6132 . . . . . . . . . 10  |-  ( ( J  Fn  ( t  X.  t )  /\  ( t  X.  t
)  e.  _V )  ->  J  e.  _V )
73, 5, 6sylancl 668 . . . . . . . . 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 6725 . . . . . . . . 9  |-  ( J  e.  _V  ->  ran  J  e.  _V )
9 uniexg 6588 . . . . . . . . 9  |-  ( ran 
J  e.  _V  ->  U.
ran  J  e.  _V )
10 pwexg 4587 . . . . . . . . 9  |-  ( U. ran  J  e.  _V  ->  ~P
U. ran  J  e.  _V )
117, 8, 9, 104syl 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 )
12 fndm 5675 . . . . . . . . . 10  |-  ( J  Fn  ( t  X.  t )  ->  dom  J  =  ( t  X.  t ) )
1312adantr 467 . . . . . . . . 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 2537 . . . . . . . 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 7535 . . . . . . . . . . 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 5888 . . . . . . . . . . . . 13  |-  ( J `
 x )  C_  U.
ran  J
17 sspwb 4649 . . . . . . . . . . . . 13  |-  ( ( J `  x ) 
C_  U. ran  J  <->  ~P ( J `  x )  C_ 
~P U. ran  J )
1816, 17mpbi 212 . . . . . . . . . . . 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 2751 . . . . . . . . . 10  |-  X_ x  e.  ( s  X.  s
) ~P ( J `
 x )  C_  X_ x  e.  ( s  X.  s ) ~P
U. ran  J
21 simprr 766 . . . . . . . . . 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 3430 . . . . . . . . 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 3048 . . . . . . . . . 10  |-  h  e. 
_V
2423elixpconst 7530 . . . . . . . . 9  |-  ( h  e.  X_ x  e.  ( s  X.  s ) ~P U. ran  J  <->  h : ( s  X.  s ) --> ~P U. ran  J )
2522, 24sylib 200 . . . . . . . 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 3960 . . . . . . . . . . 11  |-  ( s  e.  ~P t  -> 
s  C_  t )
2726ad2antrl 734 . . . . . . . . . 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 4940 . . . . . . . . . 10  |-  ( ( s  C_  t  /\  s  C_  t )  -> 
( s  X.  s
)  C_  ( t  X.  t ) )
2927, 27, 28syl2anc 667 . . . . . . . . 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 3469 . . . . . . . 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 7489 . . . . . . . 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 1269 . . . . . . 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 2880 . . . . . 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 431 . . . . 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 1776 . . . 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 199 . . 3  |-  ( h 
C_cat  J  ->  h  e.  ( ~P U. ran  J  ^pm  dom  J ) )
3736abssi 3504 . 2  |-  { h  |  h  C_cat  J }  C_  ( ~P U. ran  J 
^pm  dom  J )
381, 37ssexi 4548 1  |-  { h  |  h  C_cat  J }  e.  _V
Colors of variables: wff setvar class
Syntax hints:    /\ wa 371    = wceq 1444   E.wex 1663    e. wcel 1887   {cab 2437   E.wrex 2738   _Vcvv 3045    C_ wss 3404   ~Pcpw 3951   U.cuni 4198   class class class wbr 4402    X. cxp 4832   dom cdm 4834   ran crn 4835    Fn wfn 5577   -->wf 5578   ` cfv 5582  (class class class)co 6290    ^pm cpm 7473   X_cixp 7522    C_cat cssc 15712
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-rep 4515  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-ral 2742  df-rex 2743  df-reu 2744  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-iun 4280  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-iota 5546  df-fun 5584  df-fn 5585  df-f 5586  df-f1 5587  df-fo 5588  df-f1o 5589  df-fv 5590  df-ov 6293  df-oprab 6294  df-mpt2 6295  df-pm 7475  df-ixp 7523  df-ssc 15715
This theorem is referenced by:  issubc  15740
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