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Theorem mrcf 14669
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 14666 . 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 14668 . . 3  |-  ( C  e.  (Moore `  X
)  ->  F  =  ( x  e.  ~P X  |->  |^| { s  e.  C  |  x  C_  s } ) )
43feq1d 5657 . 2  |-  ( C  e.  (Moore `  X
)  ->  ( F : ~P X --> C  <->  ( x  e.  ~P X  |->  |^| { s  e.  C  |  x 
C_  s } ) : ~P X --> C ) )
51, 4mpbird 232 1  |-  ( C  e.  (Moore `  X
)  ->  F : ~P X --> C )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    e. wcel 1758   {crab 2803    C_ wss 3439   ~Pcpw 3971   |^|cint 4239    |-> cmpt 4461   -->wf 5525   ` cfv 5529  Moorecmre 14642  mrClscmrc 14643
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-int 4240  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-fv 5537  df-mre 14646  df-mrc 14647
This theorem is referenced by:  mrccl  14671  mrcssv  14674  mrcuni  14681  mrcun  14682  isacs2  14713  isacs4lem  15460  isacs5  15464  ismrcd2  29203  ismrc  29205  isnacs2  29210  isnacs3  29214
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