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Theorem flimclsi 20347
Description: The convergent points of a filter are a subset of the closure of any of the filter sets. (Contributed by Mario Carneiro, 9-Apr-2015.) (Revised by Stefan O'Rear, 9-Aug-2015.)
Assertion
Ref Expression
flimclsi  |-  ( S  e.  F  ->  ( J  fLim  F )  C_  ( ( cls `  J
) `  S )
)

Proof of Theorem flimclsi
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2467 . . . . . . . 8  |-  U. J  =  U. J
21flimfil 20338 . . . . . . 7  |-  ( x  e.  ( J  fLim  F )  ->  F  e.  ( Fil `  U. J
) )
32ad2antlr 726 . . . . . 6  |-  ( ( ( S  e.  F  /\  x  e.  ( J  fLim  F ) )  /\  y  e.  ( ( nei `  J
) `  { x } ) )  ->  F  e.  ( Fil ` 
U. J ) )
4 flimnei 20336 . . . . . . 7  |-  ( ( x  e.  ( J 
fLim  F )  /\  y  e.  ( ( nei `  J
) `  { x } ) )  -> 
y  e.  F )
54adantll 713 . . . . . 6  |-  ( ( ( S  e.  F  /\  x  e.  ( J  fLim  F ) )  /\  y  e.  ( ( nei `  J
) `  { x } ) )  -> 
y  e.  F )
6 simpll 753 . . . . . 6  |-  ( ( ( S  e.  F  /\  x  e.  ( J  fLim  F ) )  /\  y  e.  ( ( nei `  J
) `  { x } ) )  ->  S  e.  F )
7 filinn0 20229 . . . . . 6  |-  ( ( F  e.  ( Fil `  U. J )  /\  y  e.  F  /\  S  e.  F )  ->  ( y  i^i  S
)  =/=  (/) )
83, 5, 6, 7syl3anc 1228 . . . . 5  |-  ( ( ( S  e.  F  /\  x  e.  ( J  fLim  F ) )  /\  y  e.  ( ( nei `  J
) `  { x } ) )  -> 
( y  i^i  S
)  =/=  (/) )
98ralrimiva 2881 . . . 4  |-  ( ( S  e.  F  /\  x  e.  ( J  fLim  F ) )  ->  A. y  e.  (
( nei `  J
) `  { x } ) ( y  i^i  S )  =/=  (/) )
10 flimtop 20334 . . . . . 6  |-  ( x  e.  ( J  fLim  F )  ->  J  e.  Top )
1110adantl 466 . . . . 5  |-  ( ( S  e.  F  /\  x  e.  ( J  fLim  F ) )  ->  J  e.  Top )
12 filelss 20221 . . . . . . 7  |-  ( ( F  e.  ( Fil `  U. J )  /\  S  e.  F )  ->  S  C_  U. J )
1312ancoms 453 . . . . . 6  |-  ( ( S  e.  F  /\  F  e.  ( Fil ` 
U. J ) )  ->  S  C_  U. J
)
142, 13sylan2 474 . . . . 5  |-  ( ( S  e.  F  /\  x  e.  ( J  fLim  F ) )  ->  S  C_  U. J )
151flimelbas 20337 . . . . . 6  |-  ( x  e.  ( J  fLim  F )  ->  x  e.  U. J )
1615adantl 466 . . . . 5  |-  ( ( S  e.  F  /\  x  e.  ( J  fLim  F ) )  ->  x  e.  U. J )
171neindisj2 19492 . . . . 5  |-  ( ( J  e.  Top  /\  S  C_  U. J  /\  x  e.  U. J )  ->  ( x  e.  ( ( cls `  J
) `  S )  <->  A. y  e.  ( ( nei `  J ) `
 { x }
) ( y  i^i 
S )  =/=  (/) ) )
1811, 14, 16, 17syl3anc 1228 . . . 4  |-  ( ( S  e.  F  /\  x  e.  ( J  fLim  F ) )  -> 
( x  e.  ( ( cls `  J
) `  S )  <->  A. y  e.  ( ( nei `  J ) `
 { x }
) ( y  i^i 
S )  =/=  (/) ) )
199, 18mpbird 232 . . 3  |-  ( ( S  e.  F  /\  x  e.  ( J  fLim  F ) )  ->  x  e.  ( ( cls `  J ) `  S ) )
2019ex 434 . 2  |-  ( S  e.  F  ->  (
x  e.  ( J 
fLim  F )  ->  x  e.  ( ( cls `  J
) `  S )
) )
2120ssrdv 3515 1  |-  ( S  e.  F  ->  ( J  fLim  F )  C_  ( ( cls `  J
) `  S )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    e. wcel 1767    =/= wne 2662   A.wral 2817    i^i cin 3480    C_ wss 3481   (/)c0 3790   {csn 4033   U.cuni 4251   ` cfv 5594  (class class class)co 6295   Topctop 19263   clsccl 19387   neicnei 19466   Filcfil 20214    fLim cflim 20303
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 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
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-nel 2665  df-ral 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-iin 4334  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-fbas 18286  df-top 19268  df-cld 19388  df-ntr 19389  df-cls 19390  df-nei 19467  df-fil 20215  df-flim 20308
This theorem is referenced by:  flimcls  20354  flimfcls  20395  cnextcn  20435  cmetss  21621  minveclem4  21715
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