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Theorem clslp 17134
Description: The closure of a subset of a topological space is the subset together with its limit points. Theorem 6.6 of [Munkres] p. 97. (Contributed by NM, 26-Feb-2007.)
Hypothesis
Ref Expression
lpfval.1  |-  X  = 
U. J
Assertion
Ref Expression
clslp  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( cls `  J
) `  S )  =  ( S  u.  ( ( limPt `  J
) `  S )
) )

Proof of Theorem clslp
Dummy variables  n  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 lpfval.1 . . . . . . . . . . . . 13  |-  X  = 
U. J
21neindisj 17104 . . . . . . . . . . . 12  |-  ( ( ( J  e.  Top  /\  S  C_  X )  /\  ( x  e.  ( ( cls `  J
) `  S )  /\  n  e.  (
( nei `  J
) `  { x } ) ) )  ->  ( n  i^i 
S )  =/=  (/) )
32expr 599 . . . . . . . . . . 11  |-  ( ( ( J  e.  Top  /\  S  C_  X )  /\  x  e.  (
( cls `  J
) `  S )
)  ->  ( n  e.  ( ( nei `  J
) `  { x } )  ->  (
n  i^i  S )  =/=  (/) ) )
43adantr 452 . . . . . . . . . 10  |-  ( ( ( ( J  e. 
Top  /\  S  C_  X
)  /\  x  e.  ( ( cls `  J
) `  S )
)  /\  -.  x  e.  S )  ->  (
n  e.  ( ( nei `  J ) `
 { x }
)  ->  ( n  i^i  S )  =/=  (/) ) )
5 difsn 3876 . . . . . . . . . . . . 13  |-  ( -.  x  e.  S  -> 
( S  \  {
x } )  =  S )
65ineq2d 3485 . . . . . . . . . . . 12  |-  ( -.  x  e.  S  -> 
( n  i^i  ( S  \  { x }
) )  =  ( n  i^i  S ) )
76neeq1d 2563 . . . . . . . . . . 11  |-  ( -.  x  e.  S  -> 
( ( n  i^i  ( S  \  {
x } ) )  =/=  (/)  <->  ( n  i^i 
S )  =/=  (/) ) )
87adantl 453 . . . . . . . . . 10  |-  ( ( ( ( J  e. 
Top  /\  S  C_  X
)  /\  x  e.  ( ( cls `  J
) `  S )
)  /\  -.  x  e.  S )  ->  (
( n  i^i  ( S  \  { x }
) )  =/=  (/)  <->  ( n  i^i  S )  =/=  (/) ) )
94, 8sylibrd 226 . . . . . . . . 9  |-  ( ( ( ( J  e. 
Top  /\  S  C_  X
)  /\  x  e.  ( ( cls `  J
) `  S )
)  /\  -.  x  e.  S )  ->  (
n  e.  ( ( nei `  J ) `
 { x }
)  ->  ( n  i^i  ( S  \  {
x } ) )  =/=  (/) ) )
109ex 424 . . . . . . . 8  |-  ( ( ( J  e.  Top  /\  S  C_  X )  /\  x  e.  (
( cls `  J
) `  S )
)  ->  ( -.  x  e.  S  ->  ( n  e.  ( ( nei `  J ) `
 { x }
)  ->  ( n  i^i  ( S  \  {
x } ) )  =/=  (/) ) ) )
1110ralrimdv 2738 . . . . . . 7  |-  ( ( ( J  e.  Top  /\  S  C_  X )  /\  x  e.  (
( cls `  J
) `  S )
)  ->  ( -.  x  e.  S  ->  A. n  e.  ( ( nei `  J ) `
 { x }
) ( n  i^i  ( S  \  {
x } ) )  =/=  (/) ) )
12 simpll 731 . . . . . . . 8  |-  ( ( ( J  e.  Top  /\  S  C_  X )  /\  x  e.  (
( cls `  J
) `  S )
)  ->  J  e.  Top )
13 simplr 732 . . . . . . . 8  |-  ( ( ( J  e.  Top  /\  S  C_  X )  /\  x  e.  (
( cls `  J
) `  S )
)  ->  S  C_  X
)
141clsss3 17046 . . . . . . . . 9  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( cls `  J
) `  S )  C_  X )
1514sselda 3291 . . . . . . . 8  |-  ( ( ( J  e.  Top  /\  S  C_  X )  /\  x  e.  (
( cls `  J
) `  S )
)  ->  x  e.  X )
161islp2 17132 . . . . . . . 8  |-  ( ( J  e.  Top  /\  S  C_  X  /\  x  e.  X )  ->  (
x  e.  ( (
limPt `  J ) `  S )  <->  A. n  e.  ( ( nei `  J
) `  { x } ) ( n  i^i  ( S  \  { x } ) )  =/=  (/) ) )
1712, 13, 15, 16syl3anc 1184 . . . . . . 7  |-  ( ( ( J  e.  Top  /\  S  C_  X )  /\  x  e.  (
( cls `  J
) `  S )
)  ->  ( x  e.  ( ( limPt `  J
) `  S )  <->  A. n  e.  ( ( nei `  J ) `
 { x }
) ( n  i^i  ( S  \  {
x } ) )  =/=  (/) ) )
1811, 17sylibrd 226 . . . . . 6  |-  ( ( ( J  e.  Top  /\  S  C_  X )  /\  x  e.  (
( cls `  J
) `  S )
)  ->  ( -.  x  e.  S  ->  x  e.  ( ( limPt `  J ) `  S
) ) )
1918orrd 368 . . . . 5  |-  ( ( ( J  e.  Top  /\  S  C_  X )  /\  x  e.  (
( cls `  J
) `  S )
)  ->  ( x  e.  S  \/  x  e.  ( ( limPt `  J
) `  S )
) )
20 elun 3431 . . . . 5  |-  ( x  e.  ( S  u.  ( ( limPt `  J
) `  S )
)  <->  ( x  e.  S  \/  x  e.  ( ( limPt `  J
) `  S )
) )
2119, 20sylibr 204 . . . 4  |-  ( ( ( J  e.  Top  /\  S  C_  X )  /\  x  e.  (
( cls `  J
) `  S )
)  ->  x  e.  ( S  u.  (
( limPt `  J ) `  S ) ) )
2221ex 424 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( x  e.  ( ( cls `  J
) `  S )  ->  x  e.  ( S  u.  ( ( limPt `  J ) `  S
) ) ) )
2322ssrdv 3297 . 2  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( cls `  J
) `  S )  C_  ( S  u.  (
( limPt `  J ) `  S ) ) )
241sscls 17043 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X )  ->  S  C_  ( ( cls `  J ) `  S
) )
251lpsscls 17128 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( limPt `  J
) `  S )  C_  ( ( cls `  J
) `  S )
)
2624, 25unssd 3466 . 2  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( S  u.  (
( limPt `  J ) `  S ) )  C_  ( ( cls `  J
) `  S )
)
2723, 26eqssd 3308 1  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( cls `  J
) `  S )  =  ( S  u.  ( ( limPt `  J
) `  S )
) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    \/ wo 358    /\ wa 359    = wceq 1649    e. wcel 1717    =/= wne 2550   A.wral 2649    \ cdif 3260    u. cun 3261    i^i cin 3262    C_ wss 3263   (/)c0 3571   {csn 3757   U.cuni 3957   ` cfv 5394   Topctop 16881   clsccl 17005   neicnei 17084   limPtclp 17121
This theorem is referenced by:  islpi  17135  cldlp  17136  perfcls  17351
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-rep 4261  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344  ax-un 4641
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-reu 2656  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-int 3993  df-iun 4037  df-iin 4038  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fn 5397  df-f 5398  df-f1 5399  df-fo 5400  df-f1o 5401  df-fv 5402  df-top 16886  df-cld 17006  df-ntr 17007  df-cls 17008  df-nei 17085  df-lp 17123
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