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Theorem fncld 13981
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 vuniex 4452 . . . 4  |-  U. j  e.  _V
21pwex 4197 . . 3  |-  ~P U. j  e.  _V
32rabex 4161 . 2  |-  { x  e.  ~P U. j  |  ( U. j  \  x )  e.  j }  e.  _V
4 df-cld 13978 . 2  |-  Clsd  =  ( j  e.  Top  |->  { x  e.  ~P U. j  |  ( U. j  \  x )  e.  j } )
53, 4fnmpti 5358 1  |-  Clsd  Fn  Top
Colors of variables: wff set class
Syntax hints:    e. wcel 2159   {crab 2471    \ cdif 3140   ~Pcpw 3589   U.cuni 3823    Fn wfn 5225   Topctop 13880   Clsdccld 13975
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2161  ax-14 2162  ax-ext 2170  ax-sep 4135  ax-pow 4188  ax-pr 4223  ax-un 4447
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-nf 1471  df-sb 1773  df-eu 2040  df-mo 2041  df-clab 2175  df-cleq 2181  df-clel 2184  df-nfc 2320  df-ral 2472  df-rex 2473  df-rab 2476  df-v 2753  df-un 3147  df-in 3149  df-ss 3156  df-pw 3591  df-sn 3612  df-pr 3613  df-op 3615  df-uni 3824  df-br 4018  df-opab 4079  df-mpt 4080  df-id 4307  df-xp 4646  df-rel 4647  df-cnv 4648  df-co 4649  df-dm 4650  df-fun 5232  df-fn 5233  df-cld 13978
This theorem is referenced by:  cldrcl  13985
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