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Theorem cldlp 17172
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 17087 . 2  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( S  e.  (
Clsd `  J )  <->  ( ( cls `  J
) `  S )  =  S ) )
31clslp 17170 . . . 4  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( cls `  J
) `  S )  =  ( S  u.  ( ( limPt `  J
) `  S )
) )
43eqeq1d 2416 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( ( cls `  J ) `  S
)  =  S  <->  ( S  u.  ( ( limPt `  J
) `  S )
)  =  S ) )
5 ssequn2 3484 . . 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 1649    e. wcel 1721    u. cun 3282    C_ wss 3284   U.cuni 3979   ` cfv 5417   Topctop 16917   Clsdccld 17039   clsccl 17041   limPtclp 17157
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2389  ax-rep 4284  ax-sep 4294  ax-nul 4302  ax-pow 4341  ax-pr 4367  ax-un 4664
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2262  df-mo 2263  df-clab 2395  df-cleq 2401  df-clel 2404  df-nfc 2533  df-ne 2573  df-ral 2675  df-rex 2676  df-reu 2677  df-rab 2679  df-v 2922  df-sbc 3126  df-csb 3216  df-dif 3287  df-un 3289  df-in 3291  df-ss 3298  df-nul 3593  df-if 3704  df-pw 3765  df-sn 3784  df-pr 3785  df-op 3787  df-uni 3980  df-int 4015  df-iun 4059  df-iin 4060  df-br 4177  df-opab 4231  df-mpt 4232  df-id 4462  df-xp 4847  df-rel 4848  df-cnv 4849  df-co 4850  df-dm 4851  df-rn 4852  df-res 4853  df-ima 4854  df-iota 5381  df-fun 5419  df-fn 5420  df-f 5421  df-f1 5422  df-fo 5423  df-f1o 5424  df-fv 5425  df-top 16922  df-cld 17042  df-ntr 17043  df-cls 17044  df-nei 17121  df-lp 17159
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