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Theorem flfneii 17938
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 16914 . . . . 5  |-  ( J  e.  Top  <->  J  e.  (TopOn `  X ) )
3 flfnei 17937 . . . . 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 1220 . . . 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 608 . . 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 977 . 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 3306 . . . . 5  |-  ( n  =  N  ->  (
( F " s
)  C_  n  <->  ( F " s )  C_  N
) )
87rexbidv 2663 . . . 4  |-  ( n  =  N  ->  ( E. s  e.  L  ( F " s ) 
C_  n  <->  E. s  e.  L  ( F " s )  C_  N
) )
98rspcv 2984 . . 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 980 . 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 15 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 177    /\ wa 359    /\ w3a 936    = wceq 1649    e. wcel 1717   A.wral 2642   E.wrex 2643    C_ wss 3256   {csn 3750   U.cuni 3950   "cima 4814   -->wf 5383   ` cfv 5387  (class class class)co 6013   Topctop 16874  TopOnctopon 16875   neicnei 17077   Filcfil 17791    fLimf cflf 17881
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 2361  ax-rep 4254  ax-sep 4264  ax-nul 4272  ax-pow 4311  ax-pr 4337  ax-un 4634
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 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-nel 2546  df-ral 2647  df-rex 2648  df-reu 2649  df-rab 2651  df-v 2894  df-sbc 3098  df-csb 3188  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-pw 3737  df-sn 3756  df-pr 3757  df-op 3759  df-uni 3951  df-iun 4030  df-br 4147  df-opab 4201  df-mpt 4202  df-id 4432  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-iota 5351  df-fun 5389  df-fn 5390  df-f 5391  df-f1 5392  df-fo 5393  df-f1o 5394  df-fv 5395  df-ov 6016  df-oprab 6017  df-mpt2 6018  df-map 6949  df-fbas 16616  df-fg 16617  df-top 16879  df-topon 16882  df-nei 17078  df-fil 17792  df-fm 17884  df-flim 17885  df-flf 17886
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