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Theorem fncld 17002
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 2895 . . . . 5  |-  j  e. 
_V
21uniex 4638 . . . 4  |-  U. j  e.  _V
32pwex 4316 . . 3  |-  ~P U. j  e.  _V
43rabex 4288 . 2  |-  { x  e.  ~P U. j  |  ( U. j  \  x )  e.  j }  e.  _V
5 df-cld 16999 . 2  |-  Clsd  =  ( j  e.  Top  |->  { x  e.  ~P U. j  |  ( U. j  \  x )  e.  j } )
64, 5fnmpti 5506 1  |-  Clsd  Fn  Top
Colors of variables: wff set class
Syntax hints:    e. wcel 1717   {crab 2646    \ cdif 3253   ~Pcpw 3735   U.cuni 3950    Fn wfn 5382   Topctop 16874   Clsdccld 16996
This theorem is referenced by:  cldrcl  17006  iscldtop  17075
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-fun 5389  df-fn 5390  df-cld 16999
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