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Theorem clsval 17089
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

Proof of Theorem clsval
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 iscld.1 . . . . 5  |-  X  = 
U. J
21clsfval 17077 . . . 4  |-  ( J  e.  Top  ->  ( cls `  J )  =  ( y  e.  ~P X  |->  |^| { x  e.  ( Clsd `  J
)  |  y  C_  x } ) )
32fveq1d 5721 . . 3  |-  ( J  e.  Top  ->  (
( cls `  J
) `  S )  =  ( ( y  e.  ~P X  |->  |^|
{ x  e.  (
Clsd `  J )  |  y  C_  x }
) `  S )
)
43adantr 452 . 2  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( cls `  J
) `  S )  =  ( ( y  e.  ~P X  |->  |^|
{ x  e.  (
Clsd `  J )  |  y  C_  x }
) `  S )
)
51topopn 16967 . . . . 5  |-  ( J  e.  Top  ->  X  e.  J )
6 elpw2g 4355 . . . . 5  |-  ( X  e.  J  ->  ( S  e.  ~P X  <->  S 
C_  X ) )
75, 6syl 16 . . . 4  |-  ( J  e.  Top  ->  ( S  e.  ~P X  <->  S 
C_  X ) )
87biimpar 472 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X )  ->  S  e.  ~P X
)
91topcld 17087 . . . . 5  |-  ( J  e.  Top  ->  X  e.  ( Clsd `  J
) )
10 sseq2 3362 . . . . . 6  |-  ( x  =  X  ->  ( S  C_  x  <->  S  C_  X
) )
1110rspcev 3044 . . . . 5  |-  ( ( X  e.  ( Clsd `  J )  /\  S  C_  X )  ->  E. x  e.  ( Clsd `  J
) S  C_  x
)
129, 11sylan 458 . . . 4  |-  ( ( J  e.  Top  /\  S  C_  X )  ->  E. x  e.  ( Clsd `  J ) S 
C_  x )
13 intexrab 4351 . . . 4  |-  ( E. x  e.  ( Clsd `  J ) S  C_  x 
<-> 
|^| { x  e.  (
Clsd `  J )  |  S  C_  x }  e.  _V )
1412, 13sylib 189 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X )  ->  |^| { x  e.  (
Clsd `  J )  |  S  C_  x }  e.  _V )
15 sseq1 3361 . . . . . 6  |-  ( y  =  S  ->  (
y  C_  x  <->  S  C_  x
) )
1615rabbidv 2940 . . . . 5  |-  ( y  =  S  ->  { x  e.  ( Clsd `  J
)  |  y  C_  x }  =  {
x  e.  ( Clsd `  J )  |  S  C_  x } )
1716inteqd 4047 . . . 4  |-  ( y  =  S  ->  |^| { x  e.  ( Clsd `  J
)  |  y  C_  x }  =  |^| { x  e.  ( Clsd `  J )  |  S  C_  x } )
18 eqid 2435 . . . 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 5795 . . 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 643 . 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 2467 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 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   E.wrex 2698   {crab 2701   _Vcvv 2948    C_ wss 3312   ~Pcpw 3791   U.cuni 4007   |^|cint 4042    e. cmpt 4258   ` cfv 5445   Topctop 16946   Clsdccld 17068   clsccl 17070
This theorem is referenced by:  cldcls  17094  clscld  17099  clsf  17100  clsval2  17102  clsss  17106  sscls  17108
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-rep 4312  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-reu 2704  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-int 4043  df-iun 4087  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4875  df-rel 4876  df-cnv 4877  df-co 4878  df-dm 4879  df-rn 4880  df-res 4881  df-ima 4882  df-iota 5409  df-fun 5447  df-fn 5448  df-f 5449  df-f1 5450  df-fo 5451  df-f1o 5452  df-fv 5453  df-top 16951  df-cld 17071  df-cls 17073
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