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Theorem neiflim 20644
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 3508 . . . 4  |-  ( ( nei `  J ) `
 { A }
)  C_  ( ( nei `  J ) `  { A } )
21jctr 540 . . 3  |-  ( A  e.  X  ->  ( A  e.  X  /\  ( ( nei `  J
) `  { A } )  C_  (
( nei `  J
) `  { A } ) ) )
32adantl 464 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  X )  ->  ( A  e.  X  /\  ( ( nei `  J
) `  { A } )  C_  (
( nei `  J
) `  { A } ) ) )
4 simpl 455 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  X )  ->  J  e.  (TopOn `  X )
)
5 snssi 4160 . . . . 5  |-  ( A  e.  X  ->  { A }  C_  X )
65adantl 464 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  X )  ->  { A }  C_  X )
7 snnzg 4133 . . . . 5  |-  ( A  e.  X  ->  { A }  =/=  (/) )
87adantl 464 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  X )  ->  { A }  =/=  (/) )
9 neifil 20550 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  { A }  C_  X  /\  { A }  =/=  (/) )  -> 
( ( nei `  J
) `  { A } )  e.  ( Fil `  X ) )
104, 6, 8, 9syl3anc 1226 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  A  e.  X )  ->  (
( nei `  J
) `  { A } )  e.  ( Fil `  X ) )
11 elflim 20641 . . 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 468 . 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 367    e. wcel 1823    =/= wne 2649    C_ wss 3461   (/)c0 3783   {csn 4016   ` cfv 5570  (class class class)co 6270  TopOnctopon 19565   neicnei 19768   Filcfil 20515    fLim cflim 20604
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-rep 4550  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-nel 2652  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-csb 3421  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-iun 4317  df-br 4440  df-opab 4498  df-mpt 4499  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-ov 6273  df-oprab 6274  df-mpt2 6275  df-fbas 18614  df-top 19569  df-topon 19572  df-nei 19769  df-fil 20516  df-flim 20609
This theorem is referenced by:  flimcf  20652  cnpflf2  20670  cnpflf  20671
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