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Theorem fnmrc 15037
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 15017 . . 3  |- mrCls  =  ( c  e.  U. ran Moore  |->  ( x  e.  ~P U. c  |->  |^| { s  e.  c  |  x  C_  s } ) )
21fnmpt 5632 . 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 15031 . . 3  |-  ( c  e.  U. ran Moore  <->  c  e.  (Moore `  U. c ) )
4 mrcflem 15036 . . . . 5  |-  ( c  e.  (Moore `  U. c )  ->  (
x  e.  ~P U. c  |->  |^| { s  e.  c  |  x  C_  s } ) : ~P U. c --> c )
5 fssxp 5668 . . . . 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 3054 . . . . . . 7  |-  c  e. 
_V
87uniex 6517 . . . . . 6  |-  U. c  e.  _V
98pwex 4565 . . . . 5  |-  ~P U. c  e.  _V
109, 7xpex 6525 . . . 4  |-  ( ~P
U. c  X.  c
)  e.  _V
11 ssexg 4528 . . . 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 660 . . 3  |-  ( c  e.  (Moore `  U. c )  ->  (
x  e.  ~P U. c  |->  |^| { s  e.  c  |  x  C_  s } )  e.  _V )
133, 12sylbi 195 . 2  |-  ( c  e.  U. ran Moore  ->  (
x  e.  ~P U. c  |->  |^| { s  e.  c  |  x  C_  s } )  e.  _V )
142, 13mprg 2759 1  |- mrCls  Fn  U. ran Moore
Colors of variables: wff setvar class
Syntax hints:    e. wcel 1836   {crab 2750   _Vcvv 3051    C_ wss 3406   ~Pcpw 3944   U.cuni 4180   |^|cint 4216    |-> cmpt 4442    X. cxp 4928   ran crn 4931    Fn wfn 5508   -->wf 5509   ` cfv 5513  Moorecmre 15012  mrClscmrc 15013
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-8 1838  ax-9 1840  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2020  ax-ext 2374  ax-sep 4505  ax-nul 4513  ax-pow 4560  ax-pr 4618  ax-un 6513
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1628  df-nf 1632  df-sb 1758  df-eu 2236  df-mo 2237  df-clab 2382  df-cleq 2388  df-clel 2391  df-nfc 2546  df-ne 2593  df-ral 2751  df-rex 2752  df-rab 2755  df-v 3053  df-sbc 3270  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3729  df-if 3875  df-pw 3946  df-sn 3962  df-pr 3964  df-op 3968  df-uni 4181  df-int 4217  df-br 4385  df-opab 4443  df-mpt 4444  df-id 4726  df-xp 4936  df-rel 4937  df-cnv 4938  df-co 4939  df-dm 4940  df-rn 4941  df-res 4942  df-ima 4943  df-iota 5477  df-fun 5515  df-fn 5516  df-f 5517  df-fv 5521  df-mre 15016  df-mrc 15017
This theorem is referenced by:  ismrc  30839
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