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Theorem flfneii 17687
Description: A neighborhood of a limit point of a function contains the image of a filter element. (Contributed by Jeff Hankins, 11-Nov-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.)
Hypothesis
Ref Expression
flfneii.x  |-  X  = 
U. J
Assertion
Ref Expression
flfneii  |-  ( ( ( J  e.  Top  /\  L  e.  ( Fil `  Y )  /\  F : Y --> X )  /\  A  e.  ( ( J  fLimf  L ) `  F )  /\  N  e.  ( ( nei `  J
) `  { A } ) )  ->  E. s  e.  L  ( F " s ) 
C_  N )
Distinct variable groups:    F, s    J, s    L, s    N, s    X, s    Y, s
Allowed substitution hint:    A( s)

Proof of Theorem flfneii
Dummy variable  n is distinct from all other variables.
StepHypRef Expression
1 flfneii.x . . . . . 6  |-  X  = 
U. J
21toptopon 16671 . . . . 5  |-  ( J  e.  Top  <->  J  e.  (TopOn `  X ) )
3 flfnei 17686 . . . . 5  |-  ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
)  /\  F : Y
--> X )  ->  ( A  e.  ( ( J  fLimf  L ) `  F )  <->  ( A  e.  X  /\  A. n  e.  ( ( nei `  J
) `  { A } ) E. s  e.  L  ( F " s )  C_  n
) ) )
42, 3syl3an1b 1218 . . . 4  |-  ( ( J  e.  Top  /\  L  e.  ( Fil `  Y )  /\  F : Y --> X )  -> 
( A  e.  ( ( J  fLimf  L ) `
 F )  <->  ( A  e.  X  /\  A. n  e.  ( ( nei `  J
) `  { A } ) E. s  e.  L  ( F " s )  C_  n
) ) )
54simplbda 607 . . 3  |-  ( ( ( J  e.  Top  /\  L  e.  ( Fil `  Y )  /\  F : Y --> X )  /\  A  e.  ( ( J  fLimf  L ) `  F ) )  ->  A. n  e.  (
( nei `  J
) `  { A } ) E. s  e.  L  ( F " s )  C_  n
)
653adant3 975 . 2  |-  ( ( ( J  e.  Top  /\  L  e.  ( Fil `  Y )  /\  F : Y --> X )  /\  A  e.  ( ( J  fLimf  L ) `  F )  /\  N  e.  ( ( nei `  J
) `  { A } ) )  ->  A. n  e.  (
( nei `  J
) `  { A } ) E. s  e.  L  ( F " s )  C_  n
)
7 sseq2 3200 . . . . 5  |-  ( n  =  N  ->  (
( F " s
)  C_  n  <->  ( F " s )  C_  N
) )
87rexbidv 2564 . . . 4  |-  ( n  =  N  ->  ( E. s  e.  L  ( F " s ) 
C_  n  <->  E. s  e.  L  ( F " s )  C_  N
) )
98rspcv 2880 . . 3  |-  ( N  e.  ( ( nei `  J ) `  { A } )  ->  ( A. n  e.  (
( nei `  J
) `  { A } ) E. s  e.  L  ( F " s )  C_  n  ->  E. s  e.  L  ( F " s ) 
C_  N ) )
1093ad2ant3 978 . 2  |-  ( ( ( J  e.  Top  /\  L  e.  ( Fil `  Y )  /\  F : Y --> X )  /\  A  e.  ( ( J  fLimf  L ) `  F )  /\  N  e.  ( ( nei `  J
) `  { A } ) )  -> 
( A. n  e.  ( ( nei `  J
) `  { A } ) E. s  e.  L  ( F " s )  C_  n  ->  E. s  e.  L  ( F " s ) 
C_  N ) )
116, 10mpd 14 1  |-  ( ( ( J  e.  Top  /\  L  e.  ( Fil `  Y )  /\  F : Y --> X )  /\  A  e.  ( ( J  fLimf  L ) `  F )  /\  N  e.  ( ( nei `  J
) `  { A } ) )  ->  E. s  e.  L  ( F " s ) 
C_  N )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684   A.wral 2543   E.wrex 2544    C_ wss 3152   {csn 3640   U.cuni 3827   "cima 4692   -->wf 5251   ` cfv 5255  (class class class)co 5858   Topctop 16631  TopOnctopon 16632   neicnei 16834   Filcfil 17540    fLimf cflf 17630
This theorem is referenced by:  flfneih  24972
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-map 6774  df-top 16636  df-topon 16639  df-nei 16835  df-fbas 17520  df-fg 17521  df-fil 17541  df-fm 17633  df-flim 17634  df-flf 17635
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