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Theorem fncld 17078
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 2951 . . . . 5  |-  j  e. 
_V
21uniex 4697 . . . 4  |-  U. j  e.  _V
32pwex 4374 . . 3  |-  ~P U. j  e.  _V
43rabex 4346 . 2  |-  { x  e.  ~P U. j  |  ( U. j  \  x )  e.  j }  e.  _V
5 df-cld 17075 . 2  |-  Clsd  =  ( j  e.  Top  |->  { x  e.  ~P U. j  |  ( U. j  \  x )  e.  j } )
64, 5fnmpti 5565 1  |-  Clsd  Fn  Top
Colors of variables: wff set class
Syntax hints:    e. wcel 1725   {crab 2701    \ cdif 3309   ~Pcpw 3791   U.cuni 4007    Fn wfn 5441   Topctop 16950   Clsdccld 17072
This theorem is referenced by:  cldrcl  17082  iscldtop  17151
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-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395  ax-un 4693
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 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-fun 5448  df-fn 5449  df-cld 17075
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