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Theorem cldval 12257
Description: The set of closed sets of a topology. (Note that the set of open sets is just the topology itself, so we don't have a separate definition.) (Contributed by NM, 2-Oct-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)
Hypothesis
Ref Expression
cldval.1  |-  X  = 
U. J
Assertion
Ref Expression
cldval  |-  ( J  e.  Top  ->  ( Clsd `  J )  =  { x  e.  ~P X  |  ( X  \  x )  e.  J } )
Distinct variable groups:    x, J    x, X

Proof of Theorem cldval
Dummy variable  j is distinct from all other variables.
StepHypRef Expression
1 cldval.1 . . . 4  |-  X  = 
U. J
21topopn 12164 . . 3  |-  ( J  e.  Top  ->  X  e.  J )
3 pwexg 4099 . . 3  |-  ( X  e.  J  ->  ~P X  e.  _V )
4 rabexg 4066 . . 3  |-  ( ~P X  e.  _V  ->  { x  e.  ~P X  |  ( X  \  x )  e.  J }  e.  _V )
52, 3, 43syl 17 . 2  |-  ( J  e.  Top  ->  { x  e.  ~P X  |  ( X  \  x )  e.  J }  e.  _V )
6 unieq 3740 . . . . . 6  |-  ( j  =  J  ->  U. j  =  U. J )
76, 1syl6eqr 2188 . . . . 5  |-  ( j  =  J  ->  U. j  =  X )
87pweqd 3510 . . . 4  |-  ( j  =  J  ->  ~P U. j  =  ~P X
)
97difeq1d 3188 . . . . 5  |-  ( j  =  J  ->  ( U. j  \  x
)  =  ( X 
\  x ) )
10 eleq12 2202 . . . . 5  |-  ( ( ( U. j  \  x )  =  ( X  \  x )  /\  j  =  J )  ->  ( ( U. j  \  x
)  e.  j  <->  ( X  \  x )  e.  J
) )
119, 10mpancom 418 . . . 4  |-  ( j  =  J  ->  (
( U. j  \  x )  e.  j  <-> 
( X  \  x
)  e.  J ) )
128, 11rabeqbidv 2676 . . 3  |-  ( j  =  J  ->  { x  e.  ~P U. j  |  ( U. j  \  x )  e.  j }  =  { x  e.  ~P X  |  ( X  \  x )  e.  J } )
13 df-cld 12253 . . 3  |-  Clsd  =  ( j  e.  Top  |->  { x  e.  ~P U. j  |  ( U. j  \  x )  e.  j } )
1412, 13fvmptg 5490 . 2  |-  ( ( J  e.  Top  /\  { x  e.  ~P X  |  ( X  \  x )  e.  J }  e.  _V )  ->  ( Clsd `  J
)  =  { x  e.  ~P X  |  ( X  \  x )  e.  J } )
155, 14mpdan 417 1  |-  ( J  e.  Top  ->  ( Clsd `  J )  =  { x  e.  ~P X  |  ( X  \  x )  e.  J } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 104    = wceq 1331    e. wcel 1480   {crab 2418   _Vcvv 2681    \ cdif 3063   ~Pcpw 3505   U.cuni 3731   ` cfv 5118   Topctop 12153   Clsdccld 12250
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-rab 2423  df-v 2683  df-sbc 2905  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-mpt 3986  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-iota 5083  df-fun 5120  df-fv 5126  df-top 12154  df-cld 12253
This theorem is referenced by:  iscld  12261
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