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Theorem fnmre 13509
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 2804 . . . . 5  |-  x  e. 
_V
21pwex 4209 . . . 4  |-  ~P x  e.  _V
32pwex 4209 . . 3  |-  ~P ~P x  e.  _V
43rabex 4181 . 2  |-  { c  e.  ~P ~P x  |  ( x  e.  c  /\  A. s  e.  ~P  c ( s  =/=  (/)  ->  |^| s  e.  c ) ) }  e.  _V
5 df-mre 13504 . 2  |- Moore  =  ( x  e.  _V  |->  { c  e.  ~P ~P x  |  ( x  e.  c  /\  A. s  e.  ~P  c ( s  =/=  (/)  ->  |^| s  e.  c ) ) } )
64, 5fnmpti 5388 1  |- Moore  Fn  _V
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    e. wcel 1696    =/= wne 2459   A.wral 2556   {crab 2560   _Vcvv 2801   (/)c0 3468   ~Pcpw 3638   |^|cint 3878    Fn wfn 5266  Moorecmre 13500
This theorem is referenced by:  mreunirn  13519
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-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
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-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-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-fun 5273  df-fn 5274  df-mre 13504
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