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Theorem mrcfval 13526
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 5567 . . . . 5  |-  (Moore `  X )  C_  U. ran Moore
32sseli 3189 . . . 4  |-  ( C  e.  (Moore `  X
)  ->  C  e.  U.
ran Moore )
4 unieq 3852 . . . . . . 7  |-  ( c  =  C  ->  U. c  =  U. C )
54pweqd 3643 . . . . . 6  |-  ( c  =  C  ->  ~P U. c  =  ~P U. C )
6 rabeq 2795 . . . . . . 7  |-  ( c  =  C  ->  { s  e.  c  |  x 
C_  s }  =  { s  e.  C  |  x  C_  s } )
76inteqd 3883 . . . . . 6  |-  ( c  =  C  ->  |^| { s  e.  c  |  x 
C_  s }  =  |^| { s  e.  C  |  x  C_  s } )
85, 7mpteq12dv 4114 . . . . 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 13505 . . . . 5  |- mrCls  =  ( c  e.  U. ran Moore  |->  ( x  e.  ~P U. c  |->  |^| { s  e.  c  |  x  C_  s } ) )
10 mreunirn 13519 . . . . . . . 8  |-  ( c  e.  U. ran Moore  <->  c  e.  (Moore `  U. c ) )
11 mrcflem 13524 . . . . . . . 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 5416 . . . . . . 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 2804 . . . . . . . . 9  |-  c  e. 
_V
1615uniex 4532 . . . . . . . 8  |-  U. c  e.  _V
1716pwex 4209 . . . . . . 7  |-  ~P U. c  e.  _V
1817, 15xpex 4817 . . . . . 6  |-  ( ~P
U. c  X.  c
)  e.  _V
19 ssexg 4176 . . . . . 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 5620 . . . 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 13518 . . . . 5  |-  ( C  e.  (Moore `  X
)  ->  U. C  =  X )
2423pweqd 3643 . . . 4  |-  ( C  e.  (Moore `  X
)  ->  ~P U. C  =  ~P X )
25 eqidd 2297 . . . 4  |-  ( C  e.  (Moore `  X
)  ->  |^| { s  e.  C  |  x 
C_  s }  =  |^| { s  e.  C  |  x  C_  s } )
2624, 25mpteq12dv 4114 . . 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 2328 . 2  |-  ( C  e.  (Moore `  X
)  ->  (mrCls `  C
)  =  ( x  e.  ~P X  |->  |^|
{ s  e.  C  |  x  C_  s } ) )
281, 27syl5eq 2340 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 1632    e. wcel 1696   {crab 2560   _Vcvv 2801    C_ wss 3165   ~Pcpw 3638   U.cuni 3843   |^|cint 3878    e. cmpt 4093    X. cxp 4703   ran crn 4706   -->wf 5267   ` cfv 5271  Moorecmre 13500  mrClscmrc 13501
This theorem is referenced by:  mrcf  13527  mrcval  13528  acsficl2d  14295  mrclsp  15762  mrccls  16832
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-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-rab 2565  df-v 2803  df-sbc 3005  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-int 3879  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-fv 5279  df-mre 13504  df-mrc 13505
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