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Theorem mrcf 13527
Description: The Moore closure is a function mapping arbitrary subsets to closed sets. (Contributed by Stefan O'Rear, 31-Jan-2015.)
Hypothesis
Ref Expression
mrcfval.f  |-  F  =  (mrCls `  C )
Assertion
Ref Expression
mrcf  |-  ( C  e.  (Moore `  X
)  ->  F : ~P X --> C )

Proof of Theorem mrcf
Dummy variables  x  s are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 mrcflem 13524 . 2  |-  ( C  e.  (Moore `  X
)  ->  ( x  e.  ~P X  |->  |^| { s  e.  C  |  x 
C_  s } ) : ~P X --> C )
2 mrcfval.f . . . 4  |-  F  =  (mrCls `  C )
32mrcfval 13526 . . 3  |-  ( C  e.  (Moore `  X
)  ->  F  =  ( x  e.  ~P X  |->  |^| { s  e.  C  |  x  C_  s } ) )
43feq1d 5395 . 2  |-  ( C  e.  (Moore `  X
)  ->  ( F : ~P X --> C  <->  ( x  e.  ~P X  |->  |^| { s  e.  C  |  x 
C_  s } ) : ~P X --> C ) )
51, 4mpbird 223 1  |-  ( C  e.  (Moore `  X
)  ->  F : ~P X --> C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    e. wcel 1696   {crab 2560    C_ wss 3165   ~Pcpw 3638   |^|cint 3878    e. cmpt 4093   -->wf 5267   ` cfv 5271  Moorecmre 13500  mrClscmrc 13501
This theorem is referenced by:  mrccl  13529  mrcssv  13532  mrcuni  13539  mrcun  13540  isacs2  13571  isacs4lem  14287  isacs5  14291  ismrcd2  26877  ismrc  26879  isnacs2  26884  isnacs3  26888
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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