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Theorem mrcfval 13833
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 5754 . . . . 5  |-  (Moore `  X )  C_  U. ran Moore
32sseli 3344 . . . 4  |-  ( C  e.  (Moore `  X
)  ->  C  e.  U.
ran Moore )
4 unieq 4024 . . . . . . 7  |-  ( c  =  C  ->  U. c  =  U. C )
54pweqd 3804 . . . . . 6  |-  ( c  =  C  ->  ~P U. c  =  ~P U. C )
6 rabeq 2950 . . . . . . 7  |-  ( c  =  C  ->  { s  e.  c  |  x 
C_  s }  =  { s  e.  C  |  x  C_  s } )
76inteqd 4055 . . . . . 6  |-  ( c  =  C  ->  |^| { s  e.  c  |  x 
C_  s }  =  |^| { s  e.  C  |  x  C_  s } )
85, 7mpteq12dv 4287 . . . . 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 13812 . . . . 5  |- mrCls  =  ( c  e.  U. ran Moore  |->  ( x  e.  ~P U. c  |->  |^| { s  e.  c  |  x  C_  s } ) )
10 mreunirn 13826 . . . . . . . 8  |-  ( c  e.  U. ran Moore  <->  c  e.  (Moore `  U. c ) )
11 mrcflem 13831 . . . . . . . 8  |-  ( c  e.  (Moore `  U. c )  ->  (
x  e.  ~P U. c  |->  |^| { s  e.  c  |  x  C_  s } ) : ~P U. c --> c )
1210, 11sylbi 188 . . . . . . 7  |-  ( c  e.  U. ran Moore  ->  (
x  e.  ~P U. c  |->  |^| { s  e.  c  |  x  C_  s } ) : ~P U. c --> c )
13 fssxp 5602 . . . . . . 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 16 . . . . . 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 2959 . . . . . . . . 9  |-  c  e. 
_V
1615uniex 4705 . . . . . . . 8  |-  U. c  e.  _V
1716pwex 4382 . . . . . . 7  |-  ~P U. c  e.  _V
1817, 15xpex 4990 . . . . . 6  |-  ( ~P
U. c  X.  c
)  e.  _V
19 ssexg 4349 . . . . . 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 644 . . . . 5  |-  ( c  e.  U. ran Moore  ->  (
x  e.  ~P U. c  |->  |^| { s  e.  c  |  x  C_  s } )  e.  _V )
218, 9, 20fvmpt3 5808 . . . 4  |-  ( C  e.  U. ran Moore  ->  (mrCls `  C )  =  ( x  e.  ~P U. C  |->  |^| { s  e.  C  |  x  C_  s } ) )
223, 21syl 16 . . 3  |-  ( C  e.  (Moore `  X
)  ->  (mrCls `  C
)  =  ( x  e.  ~P U. C  |-> 
|^| { s  e.  C  |  x  C_  s } ) )
23 mreuni 13825 . . . . 5  |-  ( C  e.  (Moore `  X
)  ->  U. C  =  X )
2423pweqd 3804 . . . 4  |-  ( C  e.  (Moore `  X
)  ->  ~P U. C  =  ~P X )
2524mpteq1d 4290 . . 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 } ) )
2622, 25eqtrd 2468 . 2  |-  ( C  e.  (Moore `  X
)  ->  (mrCls `  C
)  =  ( x  e.  ~P X  |->  |^|
{ s  e.  C  |  x  C_  s } ) )
271, 26syl5eq 2480 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 1652    e. wcel 1725   {crab 2709   _Vcvv 2956    C_ wss 3320   ~Pcpw 3799   U.cuni 4015   |^|cint 4050    e. cmpt 4266    X. cxp 4876   ran crn 4879   -->wf 5450   ` cfv 5454  Moorecmre 13807  mrClscmrc 13808
This theorem is referenced by:  mrcf  13834  mrcval  13835  acsficl2d  14602  mrclsp  16065  mrccls  17143
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-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-rab 2714  df-v 2958  df-sbc 3162  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-int 4051  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-fv 5462  df-mre 13811  df-mrc 13812
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