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Theorem lpsscls 17207
Description: The limit points of a subset are included in the subset's closure. (Contributed by NM, 26-Feb-2007.)
Hypothesis
Ref Expression
lpfval.1  |-  X  = 
U. J
Assertion
Ref Expression
lpsscls  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( limPt `  J
) `  S )  C_  ( ( cls `  J
) `  S )
)

Proof of Theorem lpsscls
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 lpfval.1 . . 3  |-  X  = 
U. J
21lpval 17205 . 2  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( limPt `  J
) `  S )  =  { x  |  x  e.  ( ( cls `  J ) `  ( S  \  { x }
) ) } )
3 difss 3476 . . . . 5  |-  ( S 
\  { x }
)  C_  S
41clsss 17120 . . . . 5  |-  ( ( J  e.  Top  /\  S  C_  X  /\  ( S  \  { x }
)  C_  S )  ->  ( ( cls `  J
) `  ( S  \  { x } ) )  C_  ( ( cls `  J ) `  S ) )
53, 4mp3an3 1269 . . . 4  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( cls `  J
) `  ( S  \  { x } ) )  C_  ( ( cls `  J ) `  S ) )
65sseld 3349 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( x  e.  ( ( cls `  J
) `  ( S  \  { x } ) )  ->  x  e.  ( ( cls `  J
) `  S )
) )
76abssdv 3419 . 2  |-  ( ( J  e.  Top  /\  S  C_  X )  ->  { x  |  x  e.  ( ( cls `  J
) `  ( S  \  { x } ) ) }  C_  (
( cls `  J
) `  S )
)
82, 7eqsstrd 3384 1  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( limPt `  J
) `  S )  C_  ( ( cls `  J
) `  S )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    = wceq 1653    e. wcel 1726   {cab 2424    \ cdif 3319    C_ wss 3322   {csn 3816   U.cuni 4017   ` cfv 5456   Topctop 16960   clsccl 17084   limPtclp 17200
This theorem is referenced by:  lpss  17208  clslp  17214
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-int 4053  df-iun 4097  df-iin 4098  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-top 16965  df-cld 17085  df-cls 17087  df-lp 17202
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