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Theorem neiflim 17929
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 3312 . . . 4  |-  ( ( nei `  J ) `
 { A }
)  C_  ( ( nei `  J ) `  { A } )
21jctr 527 . . 3  |-  ( A  e.  X  ->  ( A  e.  X  /\  ( ( nei `  J
) `  { A } )  C_  (
( nei `  J
) `  { A } ) ) )
32adantl 453 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  X )  ->  ( A  e.  X  /\  ( ( nei `  J
) `  { A } )  C_  (
( nei `  J
) `  { A } ) ) )
4 simpl 444 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  X )  ->  J  e.  (TopOn `  X )
)
5 snssi 3887 . . . . 5  |-  ( A  e.  X  ->  { A }  C_  X )
65adantl 453 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  X )  ->  { A }  C_  X )
7 snnzg 3866 . . . . 5  |-  ( A  e.  X  ->  { A }  =/=  (/) )
87adantl 453 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  X )  ->  { A }  =/=  (/) )
9 neifil 17835 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  { A }  C_  X  /\  { A }  =/=  (/) )  -> 
( ( nei `  J
) `  { A } )  e.  ( Fil `  X ) )
104, 6, 8, 9syl3anc 1184 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  X )  ->  (
( nei `  J
) `  { A } )  e.  ( Fil `  X ) )
11 elflim 17926 . . 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 457 . 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 224 1  |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  X )  ->  A  e.  ( J  fLim  (
( nei `  J
) `  { A } ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    e. wcel 1717    =/= wne 2552    C_ wss 3265   (/)c0 3573   {csn 3759   ` cfv 5396  (class class class)co 6022  TopOnctopon 16884   neicnei 17086   Filcfil 17800    fLim cflim 17889
This theorem is referenced by:  flimcf  17937  cnpflf2  17955  cnpflf  17956
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2370  ax-rep 4263  ax-sep 4273  ax-nul 4281  ax-pow 4320  ax-pr 4346  ax-un 4643
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2244  df-mo 2245  df-clab 2376  df-cleq 2382  df-clel 2385  df-nfc 2514  df-ne 2554  df-nel 2555  df-ral 2656  df-rex 2657  df-reu 2658  df-rab 2660  df-v 2903  df-sbc 3107  df-csb 3197  df-dif 3268  df-un 3270  df-in 3272  df-ss 3279  df-nul 3574  df-if 3685  df-pw 3746  df-sn 3765  df-pr 3766  df-op 3768  df-uni 3960  df-iun 4039  df-br 4156  df-opab 4210  df-mpt 4211  df-id 4441  df-xp 4826  df-rel 4827  df-cnv 4828  df-co 4829  df-dm 4830  df-rn 4831  df-res 4832  df-ima 4833  df-iota 5360  df-fun 5398  df-fn 5399  df-f 5400  df-f1 5401  df-fo 5402  df-f1o 5403  df-fv 5404  df-ov 6025  df-oprab 6026  df-mpt2 6027  df-fbas 16625  df-top 16888  df-topon 16891  df-nei 17087  df-fil 17801  df-flim 17894
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