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Theorem flfneii 17703
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 16687 . . . . 5  |-  ( J  e.  Top  <->  J  e.  (TopOn `  X ) )
3 flfnei 17702 . . . . 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 3213 . . . . 5  |-  ( n  =  N  ->  (
( F " s
)  C_  n  <->  ( F " s )  C_  N
) )
87rexbidv 2577 . . . 4  |-  ( n  =  N  ->  ( E. s  e.  L  ( F " s ) 
C_  n  <->  E. s  e.  L  ( F " s )  C_  N
) )
98rspcv 2893 . . 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 1632    e. wcel 1696   A.wral 2556   E.wrex 2557    C_ wss 3165   {csn 3653   U.cuni 3843   "cima 4708   -->wf 5267   ` cfv 5271  (class class class)co 5874   Topctop 16647  TopOnctopon 16648   neicnei 16850   Filcfil 17556    fLimf cflf 17646
This theorem is referenced by:  flfneih  25663
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-map 6790  df-top 16652  df-topon 16655  df-nei 16851  df-fbas 17536  df-fg 17537  df-fil 17557  df-fm 17649  df-flim 17650  df-flf 17651
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