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

Theorem fnmre 13493
Description: The Moore collection generator is a well-behaved function. (Contributed by Stefan O'Rear, 30-Jan-2015.)
Assertion
Ref Expression
fnmre  |- Moore  Fn  _V

Proof of Theorem fnmre
Dummy variables  c 
s  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 2791 . . . . 5  |-  x  e. 
_V
21pwex 4193 . . . 4  |-  ~P x  e.  _V
32pwex 4193 . . 3  |-  ~P ~P x  e.  _V
43rabex 4165 . 2  |-  { c  e.  ~P ~P x  |  ( x  e.  c  /\  A. s  e.  ~P  c ( s  =/=  (/)  ->  |^| s  e.  c ) ) }  e.  _V
5 df-mre 13488 . 2  |- Moore  =  ( x  e.  _V  |->  { c  e.  ~P ~P x  |  ( x  e.  c  /\  A. s  e.  ~P  c ( s  =/=  (/)  ->  |^| s  e.  c ) ) } )
64, 5fnmpti 5372 1  |- Moore  Fn  _V
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    e. wcel 1684    =/= wne 2446   A.wral 2543   {crab 2547   _Vcvv 2788   (/)c0 3455   ~Pcpw 3625   |^|cint 3862    Fn wfn 5250  Moorecmre 13484
This theorem is referenced by:  mreunirn  13503
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-fun 5257  df-fn 5258  df-mre 13488
  Copyright terms: Public domain W3C validator