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Theorem cldval 18602
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 18494 . . 3  |-  ( J  e.  Top  ->  X  e.  J )
3 pwexg 4471 . . 3  |-  ( X  e.  J  ->  ~P X  e.  _V )
4 rabexg 4437 . . 3  |-  ( ~P X  e.  _V  ->  { x  e.  ~P X  |  ( X  \  x )  e.  J }  e.  _V )
52, 3, 43syl 20 . 2  |-  ( J  e.  Top  ->  { x  e.  ~P X  |  ( X  \  x )  e.  J }  e.  _V )
6 unieq 4094 . . . . . 6  |-  ( j  =  J  ->  U. j  =  U. J )
76, 1syl6eqr 2488 . . . . 5  |-  ( j  =  J  ->  U. j  =  X )
87pweqd 3860 . . . 4  |-  ( j  =  J  ->  ~P U. j  =  ~P X
)
97difeq1d 3468 . . . . 5  |-  ( j  =  J  ->  ( U. j  \  x
)  =  ( X 
\  x ) )
10 eleq12 2500 . . . . 5  |-  ( ( ( U. j  \  x )  =  ( X  \  x )  /\  j  =  J )  ->  ( ( U. j  \  x
)  e.  j  <->  ( X  \  x )  e.  J
) )
119, 10mpancom 669 . . . 4  |-  ( j  =  J  ->  (
( U. j  \  x )  e.  j  <-> 
( X  \  x
)  e.  J ) )
128, 11rabeqbidv 2962 . . 3  |-  ( j  =  J  ->  { x  e.  ~P U. j  |  ( U. j  \  x )  e.  j }  =  { x  e.  ~P X  |  ( X  \  x )  e.  J } )
13 df-cld 18598 . . 3  |-  Clsd  =  ( j  e.  Top  |->  { x  e.  ~P U. j  |  ( U. j  \  x )  e.  j } )
1412, 13fvmptg 5767 . 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 668 1  |-  ( J  e.  Top  ->  ( Clsd `  J )  =  { x  e.  ~P X  |  ( X  \  x )  e.  J } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1369    e. wcel 1756   {crab 2714   _Vcvv 2967    \ cdif 3320   ~Pcpw 3855   U.cuni 4086   ` cfv 5413   Topctop 18473   Clsdccld 18595
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-rab 2719  df-v 2969  df-sbc 3182  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-iota 5376  df-fun 5415  df-fv 5421  df-top 18478  df-cld 18598
This theorem is referenced by:  iscld  18606  mretopd  18671
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