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Theorem mrcfval 13510
Description: Value of the function expression for the Moore closure. (Contributed by Stefan O'Rear, 31-Jan-2015.)
Hypothesis
Ref Expression
mrcfval.f  |-  F  =  (mrCls `  C )
Assertion
Ref Expression
mrcfval  |-  ( C  e.  (Moore `  X
)  ->  F  =  ( x  e.  ~P X  |->  |^| { s  e.  C  |  x  C_  s } ) )
Distinct variable groups:    x, F, s    x, C, s    x, X, s

Proof of Theorem mrcfval
Dummy variable  c is distinct from all other variables.
StepHypRef Expression
1 mrcfval.f . 2  |-  F  =  (mrCls `  C )
2 fvssunirn 5551 . . . . 5  |-  (Moore `  X )  C_  U. ran Moore
32sseli 3176 . . . 4  |-  ( C  e.  (Moore `  X
)  ->  C  e.  U.
ran Moore )
4 unieq 3836 . . . . . . 7  |-  ( c  =  C  ->  U. c  =  U. C )
54pweqd 3630 . . . . . 6  |-  ( c  =  C  ->  ~P U. c  =  ~P U. C )
6 rabeq 2782 . . . . . . 7  |-  ( c  =  C  ->  { s  e.  c  |  x 
C_  s }  =  { s  e.  C  |  x  C_  s } )
76inteqd 3867 . . . . . 6  |-  ( c  =  C  ->  |^| { s  e.  c  |  x 
C_  s }  =  |^| { s  e.  C  |  x  C_  s } )
85, 7mpteq12dv 4098 . . . . 5  |-  ( c  =  C  ->  (
x  e.  ~P U. c  |->  |^| { s  e.  c  |  x  C_  s } )  =  ( x  e.  ~P U. C  |->  |^| { s  e.  C  |  x  C_  s } ) )
9 df-mrc 13489 . . . . 5  |- mrCls  =  ( c  e.  U. ran Moore  |->  ( x  e.  ~P U. c  |->  |^| { s  e.  c  |  x  C_  s } ) )
10 mreunirn 13503 . . . . . . . 8  |-  ( c  e.  U. ran Moore  <->  c  e.  (Moore `  U. c ) )
11 mrcflem 13508 . . . . . . . 8  |-  ( c  e.  (Moore `  U. c )  ->  (
x  e.  ~P U. c  |->  |^| { s  e.  c  |  x  C_  s } ) : ~P U. c --> c )
1210, 11sylbi 187 . . . . . . 7  |-  ( c  e.  U. ran Moore  ->  (
x  e.  ~P U. c  |->  |^| { s  e.  c  |  x  C_  s } ) : ~P U. c --> c )
13 fssxp 5400 . . . . . . 7  |-  ( ( x  e.  ~P U. c  |->  |^| { s  e.  c  |  x  C_  s } ) : ~P U. c --> c  ->  (
x  e.  ~P U. c  |->  |^| { s  e.  c  |  x  C_  s } )  C_  ( ~P U. c  X.  c
) )
1412, 13syl 15 . . . . . 6  |-  ( c  e.  U. ran Moore  ->  (
x  e.  ~P U. c  |->  |^| { s  e.  c  |  x  C_  s } )  C_  ( ~P U. c  X.  c
) )
15 vex 2791 . . . . . . . . 9  |-  c  e. 
_V
1615uniex 4516 . . . . . . . 8  |-  U. c  e.  _V
1716pwex 4193 . . . . . . 7  |-  ~P U. c  e.  _V
1817, 15xpex 4801 . . . . . 6  |-  ( ~P
U. c  X.  c
)  e.  _V
19 ssexg 4160 . . . . . 6  |-  ( ( ( x  e.  ~P U. c  |->  |^| { s  e.  c  |  x  C_  s } )  C_  ( ~P U. c  X.  c
)  /\  ( ~P U. c  X.  c )  e.  _V )  -> 
( x  e.  ~P U. c  |->  |^| { s  e.  c  |  x  C_  s } )  e.  _V )
2014, 18, 19sylancl 643 . . . . 5  |-  ( c  e.  U. ran Moore  ->  (
x  e.  ~P U. c  |->  |^| { s  e.  c  |  x  C_  s } )  e.  _V )
218, 9, 20fvmpt3 5604 . . . 4  |-  ( C  e.  U. ran Moore  ->  (mrCls `  C )  =  ( x  e.  ~P U. C  |->  |^| { s  e.  C  |  x  C_  s } ) )
223, 21syl 15 . . 3  |-  ( C  e.  (Moore `  X
)  ->  (mrCls `  C
)  =  ( x  e.  ~P U. C  |-> 
|^| { s  e.  C  |  x  C_  s } ) )
23 mreuni 13502 . . . . 5  |-  ( C  e.  (Moore `  X
)  ->  U. C  =  X )
2423pweqd 3630 . . . 4  |-  ( C  e.  (Moore `  X
)  ->  ~P U. C  =  ~P X )
25 eqidd 2284 . . . 4  |-  ( C  e.  (Moore `  X
)  ->  |^| { s  e.  C  |  x 
C_  s }  =  |^| { s  e.  C  |  x  C_  s } )
2624, 25mpteq12dv 4098 . . 3  |-  ( C  e.  (Moore `  X
)  ->  ( x  e.  ~P U. C  |->  |^|
{ s  e.  C  |  x  C_  s } )  =  ( x  e.  ~P X  |->  |^|
{ s  e.  C  |  x  C_  s } ) )
2722, 26eqtrd 2315 . 2  |-  ( C  e.  (Moore `  X
)  ->  (mrCls `  C
)  =  ( x  e.  ~P X  |->  |^|
{ s  e.  C  |  x  C_  s } ) )
281, 27syl5eq 2327 1  |-  ( C  e.  (Moore `  X
)  ->  F  =  ( x  e.  ~P X  |->  |^| { s  e.  C  |  x  C_  s } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   {crab 2547   _Vcvv 2788    C_ wss 3152   ~Pcpw 3625   U.cuni 3827   |^|cint 3862    e. cmpt 4077    X. cxp 4687   ran crn 4690   -->wf 5251   ` cfv 5255  Moorecmre 13484  mrClscmrc 13485
This theorem is referenced by:  mrcf  13511  mrcval  13512  acsficl2d  14279  mrclsp  15746  mrccls  16816
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-int 3863  df-iun 3907  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-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-mre 13488  df-mrc 13489
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