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Theorem flfneih 25560
Description: A neighborhood of the limit value of a convergent function whose values are in a Hausdorff space contains the image of a filter element. (Contributed by FL, 18-Nov-2010.) (Revised by Mario Carneiro, 6-Aug-2015.)
Hypotheses
Ref Expression
flfneih.x  |-  X  = 
U. J
flfneih.a  |-  A  = 
U. ( ( J 
fLimf  L ) `  F
)
Assertion
Ref Expression
flfneih  |-  ( ( ( J  e.  Haus  /\  L  e.  ( Fil `  Y )  /\  F : Y --> X )  /\  ( ( 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 flfneih
StepHypRef Expression
1 flfneih.x . . . 4  |-  X  = 
U. J
2 flfneih.a . . . 4  |-  A  = 
U. ( ( J 
fLimf  L ) `  F
)
31, 2limvinlv 25559 . . 3  |-  ( ( ( J  e.  Haus  /\  L  e.  ( Fil `  Y )  /\  F : Y --> X )  /\  ( ( J  fLimf  L ) `  F )  =/=  (/) )  ->  A  e.  ( ( J  fLimf  L ) `  F ) )
433adant3 975 . 2  |-  ( ( ( J  e.  Haus  /\  L  e.  ( Fil `  Y )  /\  F : Y --> X )  /\  ( ( J  fLimf  L ) `  F )  =/=  (/)  /\  N  e.  ( ( nei `  J
) `  { A } ) )  ->  A  e.  ( ( J  fLimf  L ) `  F ) )
5 haustop 17059 . . . 4  |-  ( J  e.  Haus  ->  J  e. 
Top )
653anim1i 1138 . . 3  |-  ( ( J  e.  Haus  /\  L  e.  ( Fil `  Y
)  /\  F : Y
--> X )  ->  ( J  e.  Top  /\  L  e.  ( Fil `  Y
)  /\  F : Y
--> X ) )
71flfneii 17687 . . 3  |-  ( ( ( 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 )
86, 7syl3an1 1215 . 2  |-  ( ( ( J  e.  Haus  /\  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 )
94, 8syld3an2 1229 1  |-  ( ( ( J  e.  Haus  /\  L  e.  ( Fil `  Y )  /\  F : Y --> X )  /\  ( ( 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    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   E.wrex 2544    C_ wss 3152   (/)c0 3455   {csn 3640   U.cuni 3827   "cima 4692   -->wf 5251   ` cfv 5255  (class class class)co 5858   Topctop 16631   neicnei 16834   Hauscha 17036   Filcfil 17540    fLimf cflf 17630
This theorem is referenced by:  flfnein  25621
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-suc 4398  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-1o 6479  df-map 6774  df-en 6864  df-top 16636  df-topon 16639  df-nei 16835  df-haus 17043  df-fbas 17520  df-fg 17521  df-fil 17541  df-fm 17633  df-flim 17634  df-flf 17635
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