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Theorem cldlp 18896
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 18810 . 2  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( S  e.  (
Clsd `  J )  <->  ( ( cls `  J
) `  S )  =  S ) )
31clslp 18894 . . . 4  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( cls `  J
) `  S )  =  ( S  u.  ( ( limPt `  J
) `  S )
) )
43eqeq1d 2456 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( ( cls `  J ) `  S
)  =  S  <->  ( S  u.  ( ( limPt `  J
) `  S )
)  =  S ) )
5 ssequn2 3640 . . 3  |-  ( ( ( limPt `  J ) `  S )  C_  S  <->  ( S  u.  ( (
limPt `  J ) `  S ) )  =  S )
64, 5syl6bbr 263 . 2  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( ( cls `  J ) `  S
)  =  S  <->  ( ( limPt `  J ) `  S )  C_  S
) )
72, 6bitrd 253 1  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( S  e.  (
Clsd `  J )  <->  ( ( limPt `  J ) `  S )  C_  S
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758    u. cun 3437    C_ wss 3439   U.cuni 4202   ` cfv 5529   Topctop 18640   Clsdccld 18762   clsccl 18764   limPtclp 18880
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-rep 4514  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-int 4240  df-iun 4284  df-iin 4285  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fn 5532  df-f 5533  df-f1 5534  df-fo 5535  df-f1o 5536  df-fv 5537  df-top 18645  df-cld 18765  df-ntr 18766  df-cls 18767  df-nei 18844  df-lp 18882
This theorem is referenced by: (None)
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