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Theorem cldval 14267
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 14176 . . 3  |-  ( J  e.  Top  ->  X  e.  J )
3 pwexg 4209 . . 3  |-  ( X  e.  J  ->  ~P X  e.  _V )
4 rabexg 4172 . . 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 3844 . . . . . 6  |-  ( j  =  J  ->  U. j  =  U. J )
76, 1eqtr4di 2244 . . . . 5  |-  ( j  =  J  ->  U. j  =  X )
87pweqd 3606 . . . 4  |-  ( j  =  J  ->  ~P U. j  =  ~P X
)
97difeq1d 3276 . . . . 5  |-  ( j  =  J  ->  ( U. j  \  x
)  =  ( X 
\  x ) )
10 eleq12 2258 . . . . 5  |-  ( ( ( U. j  \  x )  =  ( X  \  x )  /\  j  =  J )  ->  ( ( U. j  \  x
)  e.  j  <->  ( X  \  x )  e.  J
) )
119, 10mpancom 422 . . . 4  |-  ( j  =  J  ->  (
( U. j  \  x )  e.  j  <-> 
( X  \  x
)  e.  J ) )
128, 11rabeqbidv 2755 . . 3  |-  ( j  =  J  ->  { x  e.  ~P U. j  |  ( U. j  \  x )  e.  j }  =  { x  e.  ~P X  |  ( X  \  x )  e.  J } )
13 df-cld 14263 . . 3  |-  Clsd  =  ( j  e.  Top  |->  { x  e.  ~P U. j  |  ( U. j  \  x )  e.  j } )
1412, 13fvmptg 5633 . 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 421 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 105    = wceq 1364    e. wcel 2164   {crab 2476   _Vcvv 2760    \ cdif 3150   ~Pcpw 3601   U.cuni 3835   ` cfv 5254   Topctop 14165   Clsdccld 14260
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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-sbc 2986  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-iota 5215  df-fun 5256  df-fv 5262  df-top 14166  df-cld 14263
This theorem is referenced by:  iscld  14271
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