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Theorem cldlp 16875
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 16795 . 2  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( S  e.  (
Clsd `  J )  <->  ( ( cls `  J
) `  S )  =  S ) )
31clslp 16873 . . . 4  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( cls `  J
) `  S )  =  ( S  u.  ( ( limPt `  J
) `  S )
) )
43eqeq1d 2292 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( ( cls `  J ) `  S
)  =  S  <->  ( S  u.  ( ( limPt `  J
) `  S )
)  =  S ) )
5 ssequn2 3349 . . 3  |-  ( ( ( limPt `  J ) `  S )  C_  S  <->  ( S  u.  ( (
limPt `  J ) `  S ) )  =  S )
64, 5syl6bbr 256 . 2  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( ( cls `  J ) `  S
)  =  S  <->  ( ( limPt `  J ) `  S )  C_  S
) )
72, 6bitrd 246 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 6    <-> wb 178    /\ wa 360    = wceq 1624    e. wcel 1685    u. cun 3151    C_ wss 3153   U.cuni 3828   ` cfv 5221   Topctop 16625   Clsdccld 16747   clsccl 16749   limPtclp 16860
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 1867  ax-ext 2265  ax-rep 4132  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511
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 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-reu 2551  df-rab 2553  df-v 2791  df-sbc 2993  df-csb 3083  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-int 3864  df-iun 3908  df-iin 3909  df-br 4025  df-opab 4079  df-mpt 4080  df-id 4308  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-top 16630  df-cld 16750  df-ntr 16751  df-cls 16752  df-nei 16829  df-lp 16862
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