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Theorem clslp 18752
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 18721 . . . . . . . . . . . 12  |-  ( ( ( J  e.  Top  /\  S  C_  X )  /\  ( x  e.  ( ( cls `  J
) `  S )  /\  n  e.  (
( nei `  J
) `  { x } ) ) )  ->  ( n  i^i 
S )  =/=  (/) )
32expr 615 . . . . . . . . . . 11  |-  ( ( ( J  e.  Top  /\  S  C_  X )  /\  x  e.  (
( cls `  J
) `  S )
)  ->  ( n  e.  ( ( nei `  J
) `  { x } )  ->  (
n  i^i  S )  =/=  (/) ) )
43adantr 465 . . . . . . . . . 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 4008 . . . . . . . . . . . . 13  |-  ( -.  x  e.  S  -> 
( S  \  {
x } )  =  S )
65ineq2d 3552 . . . . . . . . . . . 12  |-  ( -.  x  e.  S  -> 
( n  i^i  ( S  \  { x }
) )  =  ( n  i^i  S ) )
76neeq1d 2621 . . . . . . . . . . 11  |-  ( -.  x  e.  S  -> 
( ( n  i^i  ( S  \  {
x } ) )  =/=  (/)  <->  ( n  i^i 
S )  =/=  (/) ) )
87adantl 466 . . . . . . . . . 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 234 . . . . . . . . 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 434 . . . . . . . 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 2805 . . . . . . 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 753 . . . . . . . 8  |-  ( ( ( J  e.  Top  /\  S  C_  X )  /\  x  e.  (
( cls `  J
) `  S )
)  ->  J  e.  Top )
13 simplr 754 . . . . . . . 8  |-  ( ( ( J  e.  Top  /\  S  C_  X )  /\  x  e.  (
( cls `  J
) `  S )
)  ->  S  C_  X
)
141clsss3 18663 . . . . . . . . 9  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( cls `  J
) `  S )  C_  X )
1514sselda 3356 . . . . . . . 8  |-  ( ( ( J  e.  Top  /\  S  C_  X )  /\  x  e.  (
( cls `  J
) `  S )
)  ->  x  e.  X )
161islp2 18749 . . . . . . . 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 1218 . . . . . . 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 234 . . . . . 6  |-  ( ( ( J  e.  Top  /\  S  C_  X )  /\  x  e.  (
( cls `  J
) `  S )
)  ->  ( -.  x  e.  S  ->  x  e.  ( ( limPt `  J ) `  S
) ) )
1918orrd 378 . . . . 5  |-  ( ( ( J  e.  Top  /\  S  C_  X )  /\  x  e.  (
( cls `  J
) `  S )
)  ->  ( x  e.  S  \/  x  e.  ( ( limPt `  J
) `  S )
) )
20 elun 3497 . . . . 5  |-  ( x  e.  ( S  u.  ( ( limPt `  J
) `  S )
)  <->  ( x  e.  S  \/  x  e.  ( ( limPt `  J
) `  S )
) )
2119, 20sylibr 212 . . . 4  |-  ( ( ( J  e.  Top  /\  S  C_  X )  /\  x  e.  (
( cls `  J
) `  S )
)  ->  x  e.  ( S  u.  (
( limPt `  J ) `  S ) ) )
2221ex 434 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( x  e.  ( ( cls `  J
) `  S )  ->  x  e.  ( S  u.  ( ( limPt `  J ) `  S
) ) ) )
2322ssrdv 3362 . 2  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( cls `  J
) `  S )  C_  ( S  u.  (
( limPt `  J ) `  S ) ) )
241sscls 18660 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X )  ->  S  C_  ( ( cls `  J ) `  S
) )
251lpsscls 18745 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( limPt `  J
) `  S )  C_  ( ( cls `  J
) `  S )
)
2624, 25unssd 3532 . 2  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( S  u.  (
( limPt `  J ) `  S ) )  C_  ( ( cls `  J
) `  S )
)
2723, 26eqssd 3373 1  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( cls `  J
) `  S )  =  ( S  u.  ( ( limPt `  J
) `  S )
) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2606   A.wral 2715    \ cdif 3325    u. cun 3326    i^i cin 3327    C_ wss 3328   (/)c0 3637   {csn 3877   U.cuni 4091   ` cfv 5418   Topctop 18498   clsccl 18622   neicnei 18701   limPtclp 18738
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4403  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531  ax-un 6372
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-reu 2722  df-rab 2724  df-v 2974  df-sbc 3187  df-csb 3289  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-int 4129  df-iun 4173  df-iin 4174  df-br 4293  df-opab 4351  df-mpt 4352  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853  df-iota 5381  df-fun 5420  df-fn 5421  df-f 5422  df-f1 5423  df-fo 5424  df-f1o 5425  df-fv 5426  df-top 18503  df-cld 18623  df-ntr 18624  df-cls 18625  df-nei 18702  df-lp 18740
This theorem is referenced by:  islpi  18753  cldlp  18754  perfcls  18969
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