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

Theorem fnmrc 13525
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 13505 . . 3  |- mrCls  =  ( c  e.  U. ran Moore  |->  ( x  e.  ~P U. c  |->  |^| { s  e.  c  |  x  C_  s } ) )
21fnmpt 5386 . 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 13519 . . 3  |-  ( c  e.  U. ran Moore  <->  c  e.  (Moore `  U. c ) )
4 mrcflem 13524 . . . . 5  |-  ( c  e.  (Moore `  U. c )  ->  (
x  e.  ~P U. c  |->  |^| { s  e.  c  |  x  C_  s } ) : ~P U. c --> c )
5 fssxp 5416 . . . . 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 15 . . . 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 2804 . . . . . . 7  |-  c  e. 
_V
87uniex 4532 . . . . . 6  |-  U. c  e.  _V
98pwex 4209 . . . . 5  |-  ~P U. c  e.  _V
109, 7xpex 4817 . . . 4  |-  ( ~P
U. c  X.  c
)  e.  _V
11 ssexg 4176 . . . 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 643 . . 3  |-  ( c  e.  (Moore `  U. c )  ->  (
x  e.  ~P U. c  |->  |^| { s  e.  c  |  x  C_  s } )  e.  _V )
133, 12sylbi 187 . 2  |-  ( c  e.  U. ran Moore  ->  (
x  e.  ~P U. c  |->  |^| { s  e.  c  |  x  C_  s } )  e.  _V )
142, 13mprg 2625 1  |- mrCls  Fn  U. ran Moore
Colors of variables: wff set class
Syntax hints:    e. wcel 1696   {crab 2560   _Vcvv 2801    C_ wss 3165   ~Pcpw 3638   U.cuni 3843   |^|cint 3878    e. cmpt 4093    X. cxp 4703   ran crn 4706    Fn wfn 5266   -->wf 5267   ` cfv 5271  Moorecmre 13500  mrClscmrc 13501
This theorem is referenced by:  ismrc  26879
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-mre 13504  df-mrc 13505
  Copyright terms: Public domain W3C validator