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Theorem cldval 19691
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 19582 . . 3  |-  ( J  e.  Top  ->  X  e.  J )
3 pwexg 4621 . . 3  |-  ( X  e.  J  ->  ~P X  e.  _V )
4 rabexg 4587 . . 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 4243 . . . . . 6  |-  ( j  =  J  ->  U. j  =  U. J )
76, 1syl6eqr 2513 . . . . 5  |-  ( j  =  J  ->  U. j  =  X )
87pweqd 4004 . . . 4  |-  ( j  =  J  ->  ~P U. j  =  ~P X
)
97difeq1d 3607 . . . . 5  |-  ( j  =  J  ->  ( U. j  \  x
)  =  ( X 
\  x ) )
10 eleq12 2530 . . . . 5  |-  ( ( ( U. j  \  x )  =  ( X  \  x )  /\  j  =  J )  ->  ( ( U. j  \  x
)  e.  j  <->  ( X  \  x )  e.  J
) )
119, 10mpancom 667 . . . 4  |-  ( j  =  J  ->  (
( U. j  \  x )  e.  j  <-> 
( X  \  x
)  e.  J ) )
128, 11rabeqbidv 3101 . . 3  |-  ( j  =  J  ->  { x  e.  ~P U. j  |  ( U. j  \  x )  e.  j }  =  { x  e.  ~P X  |  ( X  \  x )  e.  J } )
13 df-cld 19687 . . 3  |-  Clsd  =  ( j  e.  Top  |->  { x  e.  ~P U. j  |  ( U. j  \  x )  e.  j } )
1412, 13fvmptg 5929 . 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 666 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 1398    e. wcel 1823   {crab 2808   _Vcvv 3106    \ cdif 3458   ~Pcpw 3999   U.cuni 4235   ` cfv 5570   Topctop 19561   Clsdccld 19684
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-iota 5534  df-fun 5572  df-fv 5578  df-top 19566  df-cld 19687
This theorem is referenced by:  iscld  19695  mretopd  19760
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