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Theorem lpsscls 19933
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 19931 . 2  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( limPt `  J
) `  S )  =  { x  |  x  e.  ( ( cls `  J ) `  ( S  \  { x }
) ) } )
3 difss 3569 . . . . 5  |-  ( S 
\  { x }
)  C_  S
41clsss 19845 . . . . 5  |-  ( ( J  e.  Top  /\  S  C_  X  /\  ( S  \  { x }
)  C_  S )  ->  ( ( cls `  J
) `  ( S  \  { x } ) )  C_  ( ( cls `  J ) `  S ) )
53, 4mp3an3 1315 . . . 4  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( ( cls `  J
) `  ( S  \  { x } ) )  C_  ( ( cls `  J ) `  S ) )
65sseld 3440 . . 3  |-  ( ( J  e.  Top  /\  S  C_  X )  -> 
( x  e.  ( ( cls `  J
) `  ( S  \  { x } ) )  ->  x  e.  ( ( cls `  J
) `  S )
) )
76abssdv 3512 . 2  |-  ( ( J  e.  Top  /\  S  C_  X )  ->  { x  |  x  e.  ( ( cls `  J
) `  ( S  \  { x } ) ) }  C_  (
( cls `  J
) `  S )
)
82, 7eqsstrd 3475 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 367    = wceq 1405    e. wcel 1842   {cab 2387    \ cdif 3410    C_ wss 3413   {csn 3971   U.cuni 4190   ` cfv 5568   Topctop 19684   clsccl 19809   limPtclp 19926
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4506  ax-sep 4516  ax-nul 4524  ax-pow 4571  ax-pr 4629  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-reu 2760  df-rab 2762  df-v 3060  df-sbc 3277  df-csb 3373  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-pw 3956  df-sn 3972  df-pr 3974  df-op 3978  df-uni 4191  df-int 4227  df-iun 4272  df-iin 4273  df-br 4395  df-opab 4453  df-mpt 4454  df-id 4737  df-xp 4828  df-rel 4829  df-cnv 4830  df-co 4831  df-dm 4832  df-rn 4833  df-res 4834  df-ima 4835  df-iota 5532  df-fun 5570  df-fn 5571  df-f 5572  df-f1 5573  df-fo 5574  df-f1o 5575  df-fv 5576  df-top 19689  df-cld 19810  df-cls 19812  df-lp 19928
This theorem is referenced by:  lpss  19934  clslp  19940
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