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Theorem cldlp 17207
Description: A subset of a topological space is closed iff it contains all its limit points. Corollary 6.7 of [Munkres] p. 97. (Contributed by NM, 26-Feb-2007.)
Hypothesis
Ref Expression
lpfval.1  |-  X  = 
U. J
Assertion
Ref Expression
cldlp  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( S  e.  (
Clsd `  J )  <->  ( ( limPt `  J ) `  S )  C_  S
) )

Proof of Theorem cldlp
StepHypRef Expression
1 lpfval.1 . . 3  |-  X  = 
U. J
21iscld3 17121 . 2  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( S  e.  (
Clsd `  J )  <->  ( ( cls `  J
) `  S )  =  S ) )
31clslp 17205 . . . 4  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( cls `  J
) `  S )  =  ( S  u.  ( ( limPt `  J
) `  S )
) )
43eqeq1d 2444 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( ( cls `  J ) `  S
)  =  S  <->  ( S  u.  ( ( limPt `  J
) `  S )
)  =  S ) )
5 ssequn2 3513 . . 3  |-  ( ( ( limPt `  J ) `  S )  C_  S  <->  ( S  u.  ( (
limPt `  J ) `  S ) )  =  S )
64, 5syl6bbr 255 . 2  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( ( cls `  J ) `  S
)  =  S  <->  ( ( limPt `  J ) `  S )  C_  S
) )
72, 6bitrd 245 1  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( S  e.  (
Clsd `  J )  <->  ( ( limPt `  J ) `  S )  C_  S
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725    u. cun 3311    C_ wss 3313   U.cuni 4008   ` cfv 5447   Topctop 16951   Clsdccld 17073   clsccl 17075   limPtclp 17191
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4313  ax-sep 4323  ax-nul 4331  ax-pow 4370  ax-pr 4396  ax-un 4694
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2703  df-rex 2704  df-reu 2705  df-rab 2707  df-v 2951  df-sbc 3155  df-csb 3245  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-nul 3622  df-if 3733  df-pw 3794  df-sn 3813  df-pr 3814  df-op 3816  df-uni 4009  df-int 4044  df-iun 4088  df-iin 4089  df-br 4206  df-opab 4260  df-mpt 4261  df-id 4491  df-xp 4877  df-rel 4878  df-cnv 4879  df-co 4880  df-dm 4881  df-rn 4882  df-res 4883  df-ima 4884  df-iota 5411  df-fun 5449  df-fn 5450  df-f 5451  df-f1 5452  df-fo 5453  df-f1o 5454  df-fv 5455  df-top 16956  df-cld 17076  df-ntr 17077  df-cls 17078  df-nei 17155  df-lp 17193
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