ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  fncld Unicode version

Theorem fncld 12162
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 4328 . . . 4  |-  U. j  e.  _V
21pwex 4075 . . 3  |-  ~P U. j  e.  _V
32rabex 4040 . 2  |-  { x  e.  ~P U. j  |  ( U. j  \  x )  e.  j }  e.  _V
4 df-cld 12159 . 2  |-  Clsd  =  ( j  e.  Top  |->  { x  e.  ~P U. j  |  ( U. j  \  x )  e.  j } )
53, 4fnmpti 5219 1  |-  Clsd  Fn  Top
Colors of variables: wff set class
Syntax hints:    e. wcel 1463   {crab 2395    \ cdif 3036   ~Pcpw 3478   U.cuni 3704    Fn wfn 5086   Topctop 12059   Clsdccld 12156
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-pow 4066  ax-pr 4099  ax-un 4323
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-rab 2400  df-v 2660  df-un 3043  df-in 3045  df-ss 3052  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-br 3898  df-opab 3958  df-mpt 3959  df-id 4183  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-fun 5093  df-fn 5094  df-cld 12159
This theorem is referenced by:  cldrcl  12166
  Copyright terms: Public domain W3C validator