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Theorem clsval 16768
Description: The closure of a subset of a topology's base set is the intersection of all the closed sets that include it. Definition of closure of [Munkres] p. 94. (Contributed by NM, 10-Sep-2006.) (Revised by Mario Carneiro, 11-Nov-2013.)
Hypothesis
Ref Expression
iscld.1  |-  X  = 
U. J
Assertion
Ref Expression
clsval  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( cls `  J
) `  S )  =  |^| { x  e.  ( Clsd `  J
)  |  S  C_  x } )
Distinct variable groups:    x, J    x, S    x, X
Dummy variable  y is distinct from all other variables.

Proof of Theorem clsval
StepHypRef Expression
1 iscld.1 . . . . 5  |-  X  = 
U. J
21clsfval 16756 . . . 4  |-  ( J  e.  Top  ->  ( cls `  J )  =  ( y  e.  ~P X  |->  |^| { x  e.  ( Clsd `  J
)  |  y  C_  x } ) )
32fveq1d 5487 . . 3  |-  ( J  e.  Top  ->  (
( cls `  J
) `  S )  =  ( ( y  e.  ~P X  |->  |^|
{ x  e.  (
Clsd `  J )  |  y  C_  x }
) `  S )
)
43adantr 453 . 2  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( cls `  J
) `  S )  =  ( ( y  e.  ~P X  |->  |^|
{ x  e.  (
Clsd `  J )  |  y  C_  x }
) `  S )
)
51topopn 16646 . . . . 5  |-  ( J  e.  Top  ->  X  e.  J )
6 elpw2g 4168 . . . . 5  |-  ( X  e.  J  ->  ( S  e.  ~P X  <->  S 
C_  X ) )
75, 6syl 17 . . . 4  |-  ( J  e.  Top  ->  ( S  e.  ~P X  <->  S 
C_  X ) )
87biimpar 473 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X )  ->  S  e.  ~P X
)
91topcld 16766 . . . . 5  |-  ( J  e.  Top  ->  X  e.  ( Clsd `  J
) )
10 sseq2 3201 . . . . . 6  |-  ( x  =  X  ->  ( S  C_  x  <->  S  C_  X
) )
1110rspcev 2885 . . . . 5  |-  ( ( X  e.  ( Clsd `  J )  /\  S  C_  X )  ->  E. x  e.  ( Clsd `  J
) S  C_  x
)
129, 11sylan 459 . . . 4  |-  ( ( J  e.  Top  /\  S  C_  X )  ->  E. x  e.  ( Clsd `  J ) S 
C_  x )
13 intexrab 4173 . . . 4  |-  ( E. x  e.  ( Clsd `  J ) S  C_  x 
<-> 
|^| { x  e.  (
Clsd `  J )  |  S  C_  x }  e.  _V )
1412, 13sylib 190 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X )  ->  |^| { x  e.  (
Clsd `  J )  |  S  C_  x }  e.  _V )
15 sseq1 3200 . . . . . 6  |-  ( y  =  S  ->  (
y  C_  x  <->  S  C_  x
) )
1615rabbidv 2781 . . . . 5  |-  ( y  =  S  ->  { x  e.  ( Clsd `  J
)  |  y  C_  x }  =  {
x  e.  ( Clsd `  J )  |  S  C_  x } )
1716inteqd 3868 . . . 4  |-  ( y  =  S  ->  |^| { x  e.  ( Clsd `  J
)  |  y  C_  x }  =  |^| { x  e.  ( Clsd `  J )  |  S  C_  x } )
18 eqid 2284 . . . 4  |-  ( y  e.  ~P X  |->  |^|
{ x  e.  (
Clsd `  J )  |  y  C_  x }
)  =  ( y  e.  ~P X  |->  |^|
{ x  e.  (
Clsd `  J )  |  y  C_  x }
)
1917, 18fvmptg 5561 . . 3  |-  ( ( S  e.  ~P X  /\  |^| { x  e.  ( Clsd `  J
)  |  S  C_  x }  e.  _V )  ->  ( ( y  e.  ~P X  |->  |^|
{ x  e.  (
Clsd `  J )  |  y  C_  x }
) `  S )  =  |^| { x  e.  ( Clsd `  J
)  |  S  C_  x } )
208, 14, 19syl2anc 644 . 2  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( y  e. 
~P X  |->  |^| { x  e.  ( Clsd `  J
)  |  y  C_  x } ) `  S
)  =  |^| { x  e.  ( Clsd `  J
)  |  S  C_  x } )
214, 20eqtrd 2316 1  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( cls `  J
) `  S )  =  |^| { x  e.  ( Clsd `  J
)  |  S  C_  x } )
Colors of variables: wff set class
Syntax hints:    -> wi 6    <-> wb 178    /\ wa 360    = wceq 1624    e. wcel 1685   E.wrex 2545   {crab 2548   _Vcvv 2789    C_ wss 3153   ~Pcpw 3626   U.cuni 3828   |^|cint 3863    e. cmpt 4078   ` cfv 5221   Topctop 16625   Clsdccld 16747   clsccl 16749
This theorem is referenced by:  cldcls  16773  clscld  16778  clsf  16779  clsval2  16781  clsss  16785  sscls  16787
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-reu 2551  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-int 3864  df-iun 3908  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-top 16630  df-cld 16750  df-cls 16752
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