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

Theorem mrcf 13827
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 13824 . 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 13826 . . 3  |-  ( C  e.  (Moore `  X
)  ->  F  =  ( x  e.  ~P X  |->  |^| { s  e.  C  |  x  C_  s } ) )
43feq1d 5573 . 2  |-  ( C  e.  (Moore `  X
)  ->  ( F : ~P X --> C  <->  ( x  e.  ~P X  |->  |^| { s  e.  C  |  x 
C_  s } ) : ~P X --> C ) )
51, 4mpbird 224 1  |-  ( C  e.  (Moore `  X
)  ->  F : ~P X --> C )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    e. wcel 1725   {crab 2702    C_ wss 3313   ~Pcpw 3792   |^|cint 4043    e. cmpt 4259   -->wf 5443   ` cfv 5447  Moorecmre 13800  mrClscmrc 13801
This theorem is referenced by:  mrccl  13829  mrcssv  13832  mrcuni  13839  mrcun  13840  isacs2  13871  isacs4lem  14587  isacs5  14591  ismrcd2  26745  ismrc  26747  isnacs2  26752  isnacs3  26756
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 4323  ax-nul 4331  ax-pow 4370  ax-pr 4396  ax-un 4694
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 2703  df-rex 2704  df-rab 2707  df-v 2951  df-sbc 3155  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-nul 3622  df-if 3733  df-pw 3794  df-sn 3813  df-pr 3814  df-op 3816  df-uni 4009  df-int 4044  df-iun 4088  df-br 4206  df-opab 4260  df-mpt 4261  df-id 4491  df-xp 4877  df-rel 4878  df-cnv 4879  df-co 4880  df-dm 4881  df-rn 4882  df-res 4883  df-ima 4884  df-iota 5411  df-fun 5449  df-fn 5450  df-f 5451  df-fv 5455  df-mre 13804  df-mrc 13805
  Copyright terms: Public domain W3C validator