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Theorem lpss2 26468
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 3307 . . . . 5  |-  ( A 
C_  X  ->  ( A  \  { x }
)  C_  X )
3 ssdif 3311 . . . . 5  |-  ( B 
C_  A  ->  ( B  \  { x }
)  C_  ( A  \  { x } ) )
4 lpss2.1 . . . . . 6  |-  X  = 
U. J
54clsss 16791 . . . . 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 1224 . . . 4  |-  ( ( J  e.  Top  /\  A  C_  X  /\  B  C_  A )  ->  (
( cls `  J
) `  ( B  \  { x } ) )  C_  ( ( cls `  J ) `  ( A  \  { x } ) ) )
76sseld 3179 . . 3  |-  ( ( J  e.  Top  /\  A  C_  X  /\  B  C_  A )  ->  (
x  e.  ( ( cls `  J ) `
 ( B  \  { x } ) )  ->  x  e.  ( ( cls `  J
) `  ( A  \  { x } ) ) ) )
8 sstr 3187 . . . . . 6  |-  ( ( B  C_  A  /\  A  C_  X )  ->  B  C_  X )
98ancoms 439 . . . . 5  |-  ( ( A  C_  X  /\  B  C_  A )  ->  B  C_  X )
104islp 16872 . . . . 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 1147 . . 3  |-  ( ( J  e.  Top  /\  A  C_  X  /\  B  C_  A )  ->  (
x  e.  ( (
limPt `  J ) `  B )  <->  x  e.  ( ( cls `  J
) `  ( B  \  { x } ) ) ) )
134islp 16872 . . . 4  |-  ( ( J  e.  Top  /\  A  C_  X )  -> 
( x  e.  ( ( limPt `  J ) `  A )  <->  x  e.  ( ( cls `  J
) `  ( A  \  { x } ) ) ) )
14133adant3 975 . . 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 3185 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 934    = wceq 1623    e. wcel 1684    \ cdif 3149    C_ wss 3152   {csn 3640   U.cuni 3827   ` cfv 5255   Topctop 16631   clsccl 16755   limPtclp 16866
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-int 3863  df-iun 3907  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-top 16636  df-cld 16756  df-cls 16758  df-lp 16868
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