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Theorem clslp 17166
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 17136 . . . . . . . . . . . 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 3893 . . . . . . . . . . . . 13  |-  ( -.  x  e.  S  -> 
( S  \  {
x } )  =  S )
65ineq2d 3502 . . . . . . . . . . . 12  |-  ( -.  x  e.  S  -> 
( n  i^i  ( S  \  { x }
) )  =  ( n  i^i  S ) )
76neeq1d 2580 . . . . . . . . . . 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 2755 . . . . . . 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 17078 . . . . . . . . 9  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( cls `  J
) `  S )  C_  X )
1514sselda 3308 . . . . . . . 8  |-  ( ( ( J  e.  Top  /\  S  C_  X )  /\  x  e.  (
( cls `  J
) `  S )
)  ->  x  e.  X )
161islp2 17164 . . . . . . . 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 3448 . . . . 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 3314 . 2  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( cls `  J
) `  S )  C_  ( S  u.  (
( limPt `  J ) `  S ) ) )
241sscls 17075 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X )  ->  S  C_  ( ( cls `  J ) `  S
) )
251lpsscls 17160 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( limPt `  J
) `  S )  C_  ( ( cls `  J
) `  S )
)
2624, 25unssd 3483 . 2  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( S  u.  (
( limPt `  J ) `  S ) )  C_  ( ( cls `  J
) `  S )
)
2723, 26eqssd 3325 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 1721    =/= wne 2567   A.wral 2666    \ cdif 3277    u. cun 3278    i^i cin 3279    C_ wss 3280   (/)c0 3588   {csn 3774   U.cuni 3975   ` cfv 5413   Topctop 16913   clsccl 17037   neicnei 17116   limPtclp 17153
This theorem is referenced by:  islpi  17167  cldlp  17168  perfcls  17383
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-rep 4280  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363  ax-un 4660
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-reu 2673  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-int 4011  df-iun 4055  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-top 16918  df-cld 17038  df-ntr 17039  df-cls 17040  df-nei 17117  df-lp 17155
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