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Theorem lpsscls 19408
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 19406 . 2  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( limPt `  J
) `  S )  =  { x  |  x  e.  ( ( cls `  J ) `  ( S  \  { x }
) ) } )
3 difss 3631 . . . . 5  |-  ( S 
\  { x }
)  C_  S
41clsss 19321 . . . . 5  |-  ( ( J  e.  Top  /\  S  C_  X  /\  ( S  \  { x }
)  C_  S )  ->  ( ( cls `  J
) `  ( S  \  { x } ) )  C_  ( ( cls `  J ) `  S ) )
53, 4mp3an3 1313 . . . 4  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( cls `  J
) `  ( S  \  { x } ) )  C_  ( ( cls `  J ) `  S ) )
65sseld 3503 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( x  e.  ( ( cls `  J
) `  ( S  \  { x } ) )  ->  x  e.  ( ( cls `  J
) `  S )
) )
76abssdv 3574 . 2  |-  ( ( J  e.  Top  /\  S  C_  X )  ->  { x  |  x  e.  ( ( cls `  J
) `  ( S  \  { x } ) ) }  C_  (
( cls `  J
) `  S )
)
82, 7eqsstrd 3538 1  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( limPt `  J
) `  S )  C_  ( ( cls `  J
) `  S )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1379    e. wcel 1767   {cab 2452    \ cdif 3473    C_ wss 3476   {csn 4027   U.cuni 4245   ` cfv 5586   Topctop 19161   clsccl 19285   limPtclp 19401
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-iin 4328  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-top 19166  df-cld 19286  df-cls 19288  df-lp 19403
This theorem is referenced by:  lpss  19409  clslp  19415
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