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Theorem neiflim 17669
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 3197 . . . 4  |-  ( ( nei `  J ) `
 { A }
)  C_  ( ( nei `  J ) `  { A } )
21jctr 526 . . 3  |-  ( A  e.  X  ->  ( A  e.  X  /\  ( ( nei `  J
) `  { A } )  C_  (
( nei `  J
) `  { A } ) ) )
32adantl 452 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  X )  ->  ( A  e.  X  /\  ( ( nei `  J
) `  { A } )  C_  (
( nei `  J
) `  { A } ) ) )
4 simpl 443 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  X )  ->  J  e.  (TopOn `  X )
)
5 snssi 3759 . . . . 5  |-  ( A  e.  X  ->  { A }  C_  X )
65adantl 452 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  X )  ->  { A }  C_  X )
7 snnzg 3743 . . . . 5  |-  ( A  e.  X  ->  { A }  =/=  (/) )
87adantl 452 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  X )  ->  { A }  =/=  (/) )
9 neifil 17575 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  { A }  C_  X  /\  { A }  =/=  (/) )  -> 
( ( nei `  J
) `  { A } )  e.  ( Fil `  X ) )
104, 6, 8, 9syl3anc 1182 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  X )  ->  (
( nei `  J
) `  { A } )  e.  ( Fil `  X ) )
11 elflim 17666 . . 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 456 . 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 223 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 176    /\ wa 358    e. wcel 1684    =/= wne 2446    C_ wss 3152   (/)c0 3455   {csn 3640   ` cfv 5255  (class class class)co 5858  TopOnctopon 16632   neicnei 16834   Filcfil 17540    fLim cflim 17629
This theorem is referenced by:  flimcf  17677  cnpflf2  17695  cnpflf  17696  limfilnei  25561
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-nel 2449  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-iun 3907  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-ov 5861  df-oprab 5862  df-mpt2 5863  df-top 16636  df-topon 16639  df-nei 16835  df-fbas 17520  df-fil 17541  df-flim 17634
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