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Theorem clslp 16874
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 )
) )
Dummy variables  n  x are mutually distinct and distinct from all other variables.

Proof of Theorem clslp
StepHypRef Expression
1 lpfval.1 . . . . . . . . . . . . 13  |-  X  = 
U. J
21neindisj 16849 . . . . . . . . . . . 12  |-  ( ( ( J  e.  Top  /\  S  C_  X )  /\  ( x  e.  ( ( cls `  J
) `  S )  /\  n  e.  (
( nei `  J
) `  { x } ) ) )  ->  ( n  i^i 
S )  =/=  (/) )
32expr 600 . . . . . . . . . . 11  |-  ( ( ( J  e.  Top  /\  S  C_  X )  /\  x  e.  (
( cls `  J
) `  S )
)  ->  ( n  e.  ( ( nei `  J
) `  { x } )  ->  (
n  i^i  S )  =/=  (/) ) )
43adantr 453 . . . . . . . . . 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 3757 . . . . . . . . . . . . 13  |-  ( -.  x  e.  S  -> 
( S  \  {
x } )  =  S )
65ineq2d 3372 . . . . . . . . . . . 12  |-  ( -.  x  e.  S  -> 
( n  i^i  ( S  \  { x }
) )  =  ( n  i^i  S ) )
76neeq1d 2461 . . . . . . . . . . 11  |-  ( -.  x  e.  S  -> 
( ( n  i^i  ( S  \  {
x } ) )  =/=  (/)  <->  ( n  i^i 
S )  =/=  (/) ) )
87adantl 454 . . . . . . . . . 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 227 . . . . . . . . 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 425 . . . . . . . 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 2634 . . . . . . 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 732 . . . . . . . 8  |-  ( ( ( J  e.  Top  /\  S  C_  X )  /\  x  e.  (
( cls `  J
) `  S )
)  ->  J  e.  Top )
13 simplr 733 . . . . . . . 8  |-  ( ( ( J  e.  Top  /\  S  C_  X )  /\  x  e.  (
( cls `  J
) `  S )
)  ->  S  C_  X
)
141clsss3 16791 . . . . . . . . 9  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( cls `  J
) `  S )  C_  X )
1514sselda 3182 . . . . . . . 8  |-  ( ( ( J  e.  Top  /\  S  C_  X )  /\  x  e.  (
( cls `  J
) `  S )
)  ->  x  e.  X )
161islp2 16872 . . . . . . . 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 227 . . . . . 6  |-  ( ( ( J  e.  Top  /\  S  C_  X )  /\  x  e.  (
( cls `  J
) `  S )
)  ->  ( -.  x  e.  S  ->  x  e.  ( ( limPt `  J ) `  S
) ) )
1918orrd 369 . . . . 5  |-  ( ( ( J  e.  Top  /\  S  C_  X )  /\  x  e.  (
( cls `  J
) `  S )
)  ->  ( x  e.  S  \/  x  e.  ( ( limPt `  J
) `  S )
) )
20 elun 3318 . . . . 5  |-  ( x  e.  ( S  u.  ( ( limPt `  J
) `  S )
)  <->  ( x  e.  S  \/  x  e.  ( ( limPt `  J
) `  S )
) )
2119, 20sylibr 205 . . . 4  |-  ( ( ( J  e.  Top  /\  S  C_  X )  /\  x  e.  (
( cls `  J
) `  S )
)  ->  x  e.  ( S  u.  (
( limPt `  J ) `  S ) ) )
2221ex 425 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( x  e.  ( ( cls `  J
) `  S )  ->  x  e.  ( S  u.  ( ( limPt `  J ) `  S
) ) ) )
2322ssrdv 3187 . 2  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( cls `  J
) `  S )  C_  ( S  u.  (
( limPt `  J ) `  S ) ) )
241sscls 16788 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X )  ->  S  C_  ( ( cls `  J ) `  S
) )
251lpsscls 16868 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( limPt `  J
) `  S )  C_  ( ( cls `  J
) `  S )
)
2624, 25unssd 3353 . 2  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( S  u.  (
( limPt `  J ) `  S ) )  C_  ( ( cls `  J
) `  S )
)
2723, 26eqssd 3198 1  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( cls `  J
) `  S )  =  ( S  u.  ( ( limPt `  J
) `  S )
) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    <-> wb 178    \/ wo 359    /\ wa 360    = wceq 1624    e. wcel 1685    =/= wne 2448   A.wral 2545    \ cdif 3151    u. cun 3152    i^i cin 3153    C_ wss 3154   (/)c0 3457   {csn 3642   U.cuni 3829   ` cfv 5222   Topctop 16626   clsccl 16750   neicnei 16829   limPtclp 16861
This theorem is referenced by:  islpi  16875  cldlp  16876  perfcls  17088
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 1868  ax-ext 2266  ax-rep 4133  ax-sep 4143  ax-nul 4151  ax-pow 4188  ax-pr 4214  ax-un 4512
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 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-reu 2552  df-rab 2554  df-v 2792  df-sbc 2994  df-csb 3084  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-pw 3629  df-sn 3648  df-pr 3649  df-op 3651  df-uni 3830  df-int 3865  df-iun 3909  df-iin 3910  df-br 4026  df-opab 4080  df-mpt 4081  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-fun 5224  df-fn 5225  df-f 5226  df-f1 5227  df-fo 5228  df-f1o 5229  df-fv 5230  df-top 16631  df-cld 16751  df-ntr 16752  df-cls 16753  df-nei 16830  df-lp 16863
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