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Theorem lpss2 25975
Description: Limit points of a subset are limit points of the larger set. (Contributed by Jeff Madsen, 2-Sep-2009.)
Hypothesis
Ref Expression
lpss2.1  |-  X  = 
U. J
Assertion
Ref Expression
lpss2  |-  ( ( J  e.  Top  /\  A  C_  X  /\  B  C_  A )  ->  (
( limPt `  J ) `  B )  C_  (
( limPt `  J ) `  A ) )

Proof of Theorem lpss2
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 id 19 . . . . 5  |-  ( J  e.  Top  ->  J  e.  Top )
2 ssdifss 3394 . . . . 5  |-  ( A 
C_  X  ->  ( A  \  { x }
)  C_  X )
3 ssdif 3398 . . . . 5  |-  ( B 
C_  A  ->  ( B  \  { x }
)  C_  ( A  \  { x } ) )
4 lpss2.1 . . . . . 6  |-  X  = 
U. J
54clsss 17008 . . . . 5  |-  ( ( J  e.  Top  /\  ( A  \  { x } )  C_  X  /\  ( B  \  {
x } )  C_  ( A  \  { x } ) )  -> 
( ( cls `  J
) `  ( B  \  { x } ) )  C_  ( ( cls `  J ) `  ( A  \  { x } ) ) )
61, 2, 3, 5syl3an 1225 . . . 4  |-  ( ( J  e.  Top  /\  A  C_  X  /\  B  C_  A )  ->  (
( cls `  J
) `  ( B  \  { x } ) )  C_  ( ( cls `  J ) `  ( A  \  { x } ) ) )
76sseld 3265 . . 3  |-  ( ( J  e.  Top  /\  A  C_  X  /\  B  C_  A )  ->  (
x  e.  ( ( cls `  J ) `
 ( B  \  { x } ) )  ->  x  e.  ( ( cls `  J
) `  ( A  \  { x } ) ) ) )
8 sstr 3273 . . . . . 6  |-  ( ( B  C_  A  /\  A  C_  X )  ->  B  C_  X )
98ancoms 439 . . . . 5  |-  ( ( A  C_  X  /\  B  C_  A )  ->  B  C_  X )
104islp 17089 . . . . 5  |-  ( ( J  e.  Top  /\  B  C_  X )  -> 
( x  e.  ( ( limPt `  J ) `  B )  <->  x  e.  ( ( cls `  J
) `  ( B  \  { x } ) ) ) )
119, 10sylan2 460 . . . 4  |-  ( ( J  e.  Top  /\  ( A  C_  X  /\  B  C_  A ) )  ->  ( x  e.  ( ( limPt `  J
) `  B )  <->  x  e.  ( ( cls `  J ) `  ( B  \  { x }
) ) ) )
12113impb 1148 . . 3  |-  ( ( J  e.  Top  /\  A  C_  X  /\  B  C_  A )  ->  (
x  e.  ( (
limPt `  J ) `  B )  <->  x  e.  ( ( cls `  J
) `  ( B  \  { x } ) ) ) )
134islp 17089 . . . 4  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( x  e.  ( ( limPt `  J ) `  A )  <->  x  e.  ( ( cls `  J
) `  ( A  \  { x } ) ) ) )
14133adant3 976 . . 3  |-  ( ( J  e.  Top  /\  A  C_  X  /\  B  C_  A )  ->  (
x  e.  ( (
limPt `  J ) `  A )  <->  x  e.  ( ( cls `  J
) `  ( A  \  { x } ) ) ) )
157, 12, 143imtr4d 259 . 2  |-  ( ( J  e.  Top  /\  A  C_  X  /\  B  C_  A )  ->  (
x  e.  ( (
limPt `  J ) `  B )  ->  x  e.  ( ( limPt `  J
) `  A )
) )
1615ssrdv 3271 1  |-  ( ( J  e.  Top  /\  A  C_  X  /\  B  C_  A )  ->  (
( limPt `  J ) `  B )  C_  (
( limPt `  J ) `  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 935    = wceq 1647    e. wcel 1715    \ cdif 3235    C_ wss 3238   {csn 3729   U.cuni 3929   ` cfv 5358   Topctop 16848   clsccl 16972   limPtclp 17083
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-ral 2633  df-rex 2634  df-reu 2635  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-int 3965  df-iun 4009  df-iin 4010  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-top 16853  df-cld 16973  df-cls 16975  df-lp 17085
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