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Theorem fncld 16759
Description: The closed-set generator is a well-behaved function. (Contributed by Stefan O'Rear, 1-Feb-2015.)
Assertion
Ref Expression
fncld  |-  Clsd  Fn  Top

Proof of Theorem fncld
Dummy variables  x  j are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 2791 . . . . 5  |-  j  e. 
_V
21uniex 4516 . . . 4  |-  U. j  e.  _V
32pwex 4193 . . 3  |-  ~P U. j  e.  _V
43rabex 4165 . 2  |-  { x  e.  ~P U. j  |  ( U. j  \  x )  e.  j }  e.  _V
5 df-cld 16756 . 2  |-  Clsd  =  ( j  e.  Top  |->  { x  e.  ~P U. j  |  ( U. j  \  x )  e.  j } )
64, 5fnmpti 5372 1  |-  Clsd  Fn  Top
Colors of variables: wff set class
Syntax hints:    e. wcel 1684   {crab 2547    \ cdif 3149   ~Pcpw 3625   U.cuni 3827    Fn wfn 5250   Topctop 16631   Clsdccld 16753
This theorem is referenced by:  cldrcl  16763  iscldtop  16832
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-13 1686  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  ax-un 4512
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-uni 3828  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-cld 16756
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