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Theorem fnmre 13816
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 2959 . . . . 5  |-  x  e. 
_V
21pwex 4382 . . . 4  |-  ~P x  e.  _V
32pwex 4382 . . 3  |-  ~P ~P x  e.  _V
43rabex 4354 . 2  |-  { c  e.  ~P ~P x  |  ( x  e.  c  /\  A. s  e.  ~P  c ( s  =/=  (/)  ->  |^| s  e.  c ) ) }  e.  _V
5 df-mre 13811 . 2  |- Moore  =  ( x  e.  _V  |->  { c  e.  ~P ~P x  |  ( x  e.  c  /\  A. s  e.  ~P  c ( s  =/=  (/)  ->  |^| s  e.  c ) ) } )
64, 5fnmpti 5573 1  |- Moore  Fn  _V
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    e. wcel 1725    =/= wne 2599   A.wral 2705   {crab 2709   _Vcvv 2956   (/)c0 3628   ~Pcpw 3799   |^|cint 4050    Fn wfn 5449  Moorecmre 13807
This theorem is referenced by:  mreunirn  13826
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-fun 5456  df-fn 5457  df-mre 13811
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