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Theorem fnmrc 13834
Description: Moore-closure is a well-behaved function. (Contributed by Stefan O'Rear, 1-Feb-2015.)
Assertion
Ref Expression
fnmrc  |- mrCls  Fn  U. ran Moore

Proof of Theorem fnmrc
Dummy variables  c  x  s are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-mrc 13814 . . 3  |- mrCls  =  ( c  e.  U. ran Moore  |->  ( x  e.  ~P U. c  |->  |^| { s  e.  c  |  x  C_  s } ) )
21fnmpt 5573 . 2  |-  ( A. c  e.  U. ran Moore ( x  e.  ~P U. c  |-> 
|^| { s  e.  c  |  x  C_  s } )  e.  _V  -> mrCls 
Fn  U. ran Moore )
3 mreunirn 13828 . . 3  |-  ( c  e.  U. ran Moore  <->  c  e.  (Moore `  U. c ) )
4 mrcflem 13833 . . . . 5  |-  ( c  e.  (Moore `  U. c )  ->  (
x  e.  ~P U. c  |->  |^| { s  e.  c  |  x  C_  s } ) : ~P U. c --> c )
5 fssxp 5604 . . . . 5  |-  ( ( x  e.  ~P U. c  |->  |^| { s  e.  c  |  x  C_  s } ) : ~P U. c --> c  ->  (
x  e.  ~P U. c  |->  |^| { s  e.  c  |  x  C_  s } )  C_  ( ~P U. c  X.  c
) )
64, 5syl 16 . . . 4  |-  ( c  e.  (Moore `  U. c )  ->  (
x  e.  ~P U. c  |->  |^| { s  e.  c  |  x  C_  s } )  C_  ( ~P U. c  X.  c
) )
7 vex 2961 . . . . . . 7  |-  c  e. 
_V
87uniex 4707 . . . . . 6  |-  U. c  e.  _V
98pwex 4384 . . . . 5  |-  ~P U. c  e.  _V
109, 7xpex 4992 . . . 4  |-  ( ~P
U. c  X.  c
)  e.  _V
11 ssexg 4351 . . . 4  |-  ( ( ( x  e.  ~P U. c  |->  |^| { s  e.  c  |  x  C_  s } )  C_  ( ~P U. c  X.  c
)  /\  ( ~P U. c  X.  c )  e.  _V )  -> 
( x  e.  ~P U. c  |->  |^| { s  e.  c  |  x  C_  s } )  e.  _V )
126, 10, 11sylancl 645 . . 3  |-  ( c  e.  (Moore `  U. c )  ->  (
x  e.  ~P U. c  |->  |^| { s  e.  c  |  x  C_  s } )  e.  _V )
133, 12sylbi 189 . 2  |-  ( c  e.  U. ran Moore  ->  (
x  e.  ~P U. c  |->  |^| { s  e.  c  |  x  C_  s } )  e.  _V )
142, 13mprg 2777 1  |- mrCls  Fn  U. ran Moore
Colors of variables: wff set class
Syntax hints:    e. wcel 1726   {crab 2711   _Vcvv 2958    C_ wss 3322   ~Pcpw 3801   U.cuni 4017   |^|cint 4052    e. cmpt 4268    X. cxp 4878   ran crn 4881    Fn wfn 5451   -->wf 5452   ` cfv 5456  Moorecmre 13809  mrClscmrc 13810
This theorem is referenced by:  ismrc  26757
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
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 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-fv 5464  df-mre 13813  df-mrc 13814
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