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Theorem lpss2 31831
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
StepHypRef Expression
1 lpss2.1 . 2  |-  X  = 
U. J
21lpss3 20097 1  |-  ( ( J  e.  Top  /\  A  C_  X  /\  B  C_  A )  ->  (
( limPt `  J ) `  B )  C_  (
( limPt `  J ) `  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 982    = wceq 1437    e. wcel 1867    C_ wss 3433   U.cuni 4213   ` cfv 5592   Topctop 19854   limPtclp 20087
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-rep 4529  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-reu 2780  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-int 4250  df-iun 4295  df-iin 4296  df-br 4418  df-opab 4476  df-mpt 4477  df-id 4760  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-top 19858  df-cld 19971  df-cls 19973  df-lp 20089
This theorem is referenced by: (None)
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