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Theorem clslp 16895
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 16870 . . . . . . . . . . . 12  |-  ( ( ( J  e.  Top  /\  S  C_  X )  /\  ( x  e.  ( ( cls `  J
) `  S )  /\  n  e.  (
( nei `  J
) `  { x } ) ) )  ->  ( n  i^i 
S )  =/=  (/) )
32expr 598 . . . . . . . . . . 11  |-  ( ( ( J  e.  Top  /\  S  C_  X )  /\  x  e.  (
( cls `  J
) `  S )
)  ->  ( n  e.  ( ( nei `  J
) `  { x } )  ->  (
n  i^i  S )  =/=  (/) ) )
43adantr 451 . . . . . . . . . 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 3768 . . . . . . . . . . . . 13  |-  ( -.  x  e.  S  -> 
( S  \  {
x } )  =  S )
65ineq2d 3383 . . . . . . . . . . . 12  |-  ( -.  x  e.  S  -> 
( n  i^i  ( S  \  { x }
) )  =  ( n  i^i  S ) )
76neeq1d 2472 . . . . . . . . . . 11  |-  ( -.  x  e.  S  -> 
( ( n  i^i  ( S  \  {
x } ) )  =/=  (/)  <->  ( n  i^i 
S )  =/=  (/) ) )
87adantl 452 . . . . . . . . . 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 225 . . . . . . . . 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 423 . . . . . . . 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 2645 . . . . . . 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 730 . . . . . . . 8  |-  ( ( ( J  e.  Top  /\  S  C_  X )  /\  x  e.  (
( cls `  J
) `  S )
)  ->  J  e.  Top )
13 simplr 731 . . . . . . . 8  |-  ( ( ( J  e.  Top  /\  S  C_  X )  /\  x  e.  (
( cls `  J
) `  S )
)  ->  S  C_  X
)
141clsss3 16812 . . . . . . . . 9  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( cls `  J
) `  S )  C_  X )
1514sselda 3193 . . . . . . . 8  |-  ( ( ( J  e.  Top  /\  S  C_  X )  /\  x  e.  (
( cls `  J
) `  S )
)  ->  x  e.  X )
161islp2 16893 . . . . . . . 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 1182 . . . . . . 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 225 . . . . . 6  |-  ( ( ( J  e.  Top  /\  S  C_  X )  /\  x  e.  (
( cls `  J
) `  S )
)  ->  ( -.  x  e.  S  ->  x  e.  ( ( limPt `  J ) `  S
) ) )
1918orrd 367 . . . . 5  |-  ( ( ( J  e.  Top  /\  S  C_  X )  /\  x  e.  (
( cls `  J
) `  S )
)  ->  ( x  e.  S  \/  x  e.  ( ( limPt `  J
) `  S )
) )
20 elun 3329 . . . . 5  |-  ( x  e.  ( S  u.  ( ( limPt `  J
) `  S )
)  <->  ( x  e.  S  \/  x  e.  ( ( limPt `  J
) `  S )
) )
2119, 20sylibr 203 . . . 4  |-  ( ( ( J  e.  Top  /\  S  C_  X )  /\  x  e.  (
( cls `  J
) `  S )
)  ->  x  e.  ( S  u.  (
( limPt `  J ) `  S ) ) )
2221ex 423 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( x  e.  ( ( cls `  J
) `  S )  ->  x  e.  ( S  u.  ( ( limPt `  J ) `  S
) ) ) )
2322ssrdv 3198 . 2  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( cls `  J
) `  S )  C_  ( S  u.  (
( limPt `  J ) `  S ) ) )
241sscls 16809 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X )  ->  S  C_  ( ( cls `  J ) `  S
) )
251lpsscls 16889 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( limPt `  J
) `  S )  C_  ( ( cls `  J
) `  S )
)
2624, 25unssd 3364 . 2  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( S  u.  (
( limPt `  J ) `  S ) )  C_  ( ( cls `  J
) `  S )
)
2723, 26eqssd 3209 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 176    \/ wo 357    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   A.wral 2556    \ cdif 3162    u. cun 3163    i^i cin 3164    C_ wss 3165   (/)c0 3468   {csn 3653   U.cuni 3843   ` cfv 5271   Topctop 16647   clsccl 16771   neicnei 16850   limPtclp 16882
This theorem is referenced by:  islpi  16896  cldlp  16897  perfcls  17109
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-int 3879  df-iun 3923  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-top 16652  df-cld 16772  df-ntr 16773  df-cls 16774  df-nei 16851  df-lp 16884
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