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Theorem neiflim 20343
Description: A point is a limit point of its neighborhood filter. (Contributed by Jeff Hankins, 7-Sep-2009.) (Revised by Stefan O'Rear, 9-Aug-2015.)
Assertion
Ref Expression
neiflim  |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  X )  ->  A  e.  ( J  fLim  (
( nei `  J
) `  { A } ) ) )

Proof of Theorem neiflim
StepHypRef Expression
1 ssid 3528 . . . 4  |-  ( ( nei `  J ) `
 { A }
)  C_  ( ( nei `  J ) `  { A } )
21jctr 542 . . 3  |-  ( A  e.  X  ->  ( A  e.  X  /\  ( ( nei `  J
) `  { A } )  C_  (
( nei `  J
) `  { A } ) ) )
32adantl 466 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  X )  ->  ( A  e.  X  /\  ( ( nei `  J
) `  { A } )  C_  (
( nei `  J
) `  { A } ) ) )
4 simpl 457 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  X )  ->  J  e.  (TopOn `  X )
)
5 snssi 4177 . . . . 5  |-  ( A  e.  X  ->  { A }  C_  X )
65adantl 466 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  X )  ->  { A }  C_  X )
7 snnzg 4150 . . . . 5  |-  ( A  e.  X  ->  { A }  =/=  (/) )
87adantl 466 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  X )  ->  { A }  =/=  (/) )
9 neifil 20249 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  { A }  C_  X  /\  { A }  =/=  (/) )  -> 
( ( nei `  J
) `  { A } )  e.  ( Fil `  X ) )
104, 6, 8, 9syl3anc 1228 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  X )  ->  (
( nei `  J
) `  { A } )  e.  ( Fil `  X ) )
11 elflim 20340 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  (
( nei `  J
) `  { A } )  e.  ( Fil `  X ) )  ->  ( A  e.  ( J  fLim  (
( nei `  J
) `  { A } ) )  <->  ( A  e.  X  /\  (
( nei `  J
) `  { A } )  C_  (
( nei `  J
) `  { A } ) ) ) )
1210, 11syldan 470 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  X )  ->  ( A  e.  ( J  fLim  ( ( nei `  J
) `  { A } ) )  <->  ( A  e.  X  /\  (
( nei `  J
) `  { A } )  C_  (
( nei `  J
) `  { A } ) ) ) )
133, 12mpbird 232 1  |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  X )  ->  A  e.  ( J  fLim  (
( nei `  J
) `  { A } ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    e. wcel 1767    =/= wne 2662    C_ wss 3481   (/)c0 3790   {csn 4033   ` cfv 5594  (class class class)co 6295  TopOnctopon 19264   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-iun 4333  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-topon 19271  df-nei 19467  df-fil 20215  df-flim 20308
This theorem is referenced by:  flimcf  20351  cnpflf2  20369  cnpflf  20370
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