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Theorem fnmrc 14663
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 14643 . . 3  |- mrCls  =  ( c  e.  U. ran Moore  |->  ( x  e.  ~P U. c  |->  |^| { s  e.  c  |  x  C_  s } ) )
21fnmpt 5644 . 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 14657 . . 3  |-  ( c  e.  U. ran Moore  <->  c  e.  (Moore `  U. c ) )
4 mrcflem 14662 . . . . 5  |-  ( c  e.  (Moore `  U. c )  ->  (
x  e.  ~P U. c  |->  |^| { s  e.  c  |  x  C_  s } ) : ~P U. c --> c )
5 fssxp 5677 . . . . 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 3079 . . . . . . 7  |-  c  e. 
_V
87uniex 6485 . . . . . 6  |-  U. c  e.  _V
98pwex 4582 . . . . 5  |-  ~P U. c  e.  _V
109, 7xpex 6617 . . . 4  |-  ( ~P
U. c  X.  c
)  e.  _V
11 ssexg 4545 . . . 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 662 . . 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 2901 1  |- mrCls  Fn  U. ran Moore
Colors of variables: wff setvar class
Syntax hints:    e. wcel 1758   {crab 2802   _Vcvv 3076    C_ wss 3435   ~Pcpw 3967   U.cuni 4198   |^|cint 4235    |-> cmpt 4457    X. cxp 4945   ran crn 4948    Fn wfn 5520   -->wf 5521   ` cfv 5525  Moorecmre 14638  mrClscmrc 14639
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 4520  ax-nul 4528  ax-pow 4577  ax-pr 4638  ax-un 6481
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 2649  df-ral 2803  df-rex 2804  df-rab 2807  df-v 3078  df-sbc 3293  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-nul 3745  df-if 3899  df-pw 3969  df-sn 3985  df-pr 3987  df-op 3991  df-uni 4199  df-int 4236  df-br 4400  df-opab 4458  df-mpt 4459  df-id 4743  df-xp 4953  df-rel 4954  df-cnv 4955  df-co 4956  df-dm 4957  df-rn 4958  df-res 4959  df-ima 4960  df-iota 5488  df-fun 5527  df-fn 5528  df-f 5529  df-fv 5533  df-mre 14642  df-mrc 14643
This theorem is referenced by:  ismrc  29184
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