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Theorem flfneii 20320
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 19241 . . . . 5  |-  ( J  e.  Top  <->  J  e.  (TopOn `  X ) )
3 flfnei 20319 . . . . 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 1264 . . . 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 624 . . 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 1016 . 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 3526 . . . . 5  |-  ( n  =  N  ->  (
( F " s
)  C_  n  <->  ( F " s )  C_  N
) )
87rexbidv 2973 . . . 4  |-  ( n  =  N  ->  ( E. s  e.  L  ( F " s ) 
C_  n  <->  E. s  e.  L  ( F " s )  C_  N
) )
98rspcv 3210 . . 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 1019 . 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   A.wral 2814   E.wrex 2815    C_ wss 3476   {csn 4027   U.cuni 4245   "cima 5002   -->wf 5584   ` cfv 5588  (class class class)co 6285   Topctop 19201  TopOnctopon 19202   neicnei 19404   Filcfil 20173    fLimf cflf 20263
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-ov 6288  df-oprab 6289  df-mpt2 6290  df-map 7423  df-fbas 18227  df-fg 18228  df-top 19206  df-topon 19209  df-nei 19405  df-fil 20174  df-fm 20266  df-flim 20267  df-flf 20268
This theorem is referenced by: (None)
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