MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  mrcf Structured version   Unicode version

Theorem mrcf 13839
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 13836 . 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 13838 . . 3  |-  ( C  e.  (Moore `  X
)  ->  F  =  ( x  e.  ~P X  |->  |^| { s  e.  C  |  x  C_  s } ) )
43feq1d 5583 . 2  |-  ( C  e.  (Moore `  X
)  ->  ( F : ~P X --> C  <->  ( x  e.  ~P X  |->  |^| { s  e.  C  |  x 
C_  s } ) : ~P X --> C ) )
51, 4mpbird 225 1  |-  ( C  e.  (Moore `  X
)  ->  F : ~P X --> C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653    e. wcel 1726   {crab 2711    C_ wss 3322   ~Pcpw 3801   |^|cint 4052    e. cmpt 4269   -->wf 5453   ` cfv 5457  Moorecmre 13812  mrClscmrc 13813
This theorem is referenced by:  mrccl  13841  mrcssv  13844  mrcuni  13851  mrcun  13852  isacs2  13883  isacs4lem  14599  isacs5  14603  ismrcd2  26767  ismrc  26769  isnacs2  26774  isnacs3  26778
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-fv 5465  df-mre 13816  df-mrc 13817
  Copyright terms: Public domain W3C validator