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Theorem mrcf 15113
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 15110 . 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 15112 . . 3  |-  ( C  e.  (Moore `  X
)  ->  F  =  ( x  e.  ~P X  |->  |^| { s  e.  C  |  x  C_  s } ) )
43feq1d 5654 . 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 1403    e. wcel 1840   {crab 2755    C_ wss 3411   ~Pcpw 3952   |^|cint 4224    |-> cmpt 4450   -->wf 5519   ` cfv 5523  Moorecmre 15086  mrClscmrc 15087
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1637  ax-4 1650  ax-5 1723  ax-6 1769  ax-7 1812  ax-8 1842  ax-9 1844  ax-10 1859  ax-11 1864  ax-12 1876  ax-13 2024  ax-ext 2378  ax-sep 4514  ax-nul 4522  ax-pow 4569  ax-pr 4627  ax-un 6528
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 974  df-tru 1406  df-ex 1632  df-nf 1636  df-sb 1762  df-eu 2240  df-mo 2241  df-clab 2386  df-cleq 2392  df-clel 2395  df-nfc 2550  df-ne 2598  df-ral 2756  df-rex 2757  df-rab 2760  df-v 3058  df-sbc 3275  df-dif 3414  df-un 3416  df-in 3418  df-ss 3425  df-nul 3736  df-if 3883  df-pw 3954  df-sn 3970  df-pr 3972  df-op 3976  df-uni 4189  df-int 4225  df-br 4393  df-opab 4451  df-mpt 4452  df-id 4735  df-xp 4946  df-rel 4947  df-cnv 4948  df-co 4949  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5487  df-fun 5525  df-fn 5526  df-f 5527  df-fv 5531  df-mre 15090  df-mrc 15091
This theorem is referenced by:  mrccl  15115  mrcssv  15118  mrcuni  15125  mrcun  15126  isacs2  15157  isacs4lem  16012  isacs5  16016  ismrcd2  34957  ismrc  34959  isnacs2  34964  isnacs3  34968
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