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Theorem fcfneii 17732
Description: A neighborhood of a cluster point of a function contains a function value from every tail. (Contributed by Jeff Hankins, 27-Nov-2009.) (Revised by Stefan O'Rear, 9-Aug-2015.)
Assertion
Ref Expression
fcfneii  |-  ( ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
)  /\  F : Y
--> X )  /\  ( A  e.  ( ( J  fClusf  L ) `  F )  /\  N  e.  ( ( nei `  J
) `  { A } )  /\  S  e.  L ) )  -> 
( N  i^i  ( F " S ) )  =/=  (/) )

Proof of Theorem fcfneii
Dummy variables  n  s are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fcfnei 17730 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
)  /\  F : Y
--> X )  ->  ( A  e.  ( ( J  fClusf  L ) `  F )  <->  ( A  e.  X  /\  A. n  e.  ( ( nei `  J
) `  { A } ) A. s  e.  L  ( n  i^i  ( F " s
) )  =/=  (/) ) ) )
2 ineq1 3363 . . . . . . . 8  |-  ( n  =  N  ->  (
n  i^i  ( F " s ) )  =  ( N  i^i  ( F " s ) ) )
32neeq1d 2459 . . . . . . 7  |-  ( n  =  N  ->  (
( n  i^i  ( F " s ) )  =/=  (/)  <->  ( N  i^i  ( F " s ) )  =/=  (/) ) )
4 imaeq2 5008 . . . . . . . . 9  |-  ( s  =  S  ->  ( F " s )  =  ( F " S
) )
54ineq2d 3370 . . . . . . . 8  |-  ( s  =  S  ->  ( N  i^i  ( F "
s ) )  =  ( N  i^i  ( F " S ) ) )
65neeq1d 2459 . . . . . . 7  |-  ( s  =  S  ->  (
( N  i^i  ( F " s ) )  =/=  (/)  <->  ( N  i^i  ( F " S ) )  =/=  (/) ) )
73, 6rspc2v 2890 . . . . . 6  |-  ( ( N  e.  ( ( nei `  J ) `
 { A }
)  /\  S  e.  L )  ->  ( A. n  e.  (
( nei `  J
) `  { A } ) A. s  e.  L  ( n  i^i  ( F " s
) )  =/=  (/)  ->  ( N  i^i  ( F " S ) )  =/=  (/) ) )
87ex 423 . . . . 5  |-  ( N  e.  ( ( nei `  J ) `  { A } )  ->  ( S  e.  L  ->  ( A. n  e.  ( ( nei `  J
) `  { A } ) A. s  e.  L  ( n  i^i  ( F " s
) )  =/=  (/)  ->  ( N  i^i  ( F " S ) )  =/=  (/) ) ) )
98com3r 73 . . . 4  |-  ( A. n  e.  ( ( nei `  J ) `  { A } ) A. s  e.  L  (
n  i^i  ( F " s ) )  =/=  (/)  ->  ( N  e.  ( ( nei `  J
) `  { A } )  ->  ( S  e.  L  ->  ( N  i^i  ( F
" S ) )  =/=  (/) ) ) )
109adantl 452 . . 3  |-  ( ( A  e.  X  /\  A. n  e.  ( ( nei `  J ) `
 { A }
) A. s  e.  L  ( n  i^i  ( F " s
) )  =/=  (/) )  -> 
( N  e.  ( ( nei `  J
) `  { A } )  ->  ( S  e.  L  ->  ( N  i^i  ( F
" S ) )  =/=  (/) ) ) )
111, 10syl6bi 219 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
)  /\  F : Y
--> X )  ->  ( A  e.  ( ( J  fClusf  L ) `  F )  ->  ( N  e.  ( ( nei `  J ) `  { A } )  -> 
( S  e.  L  ->  ( N  i^i  ( F " S ) )  =/=  (/) ) ) ) )
12113imp2 1166 1  |-  ( ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
)  /\  F : Y
--> X )  /\  ( A  e.  ( ( J  fClusf  L ) `  F )  /\  N  e.  ( ( nei `  J
) `  { A } )  /\  S  e.  L ) )  -> 
( N  i^i  ( F " S ) )  =/=  (/) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    /\ w3a 934    = wceq 1623    e. wcel 1684    =/= wne 2446   A.wral 2543    i^i cin 3151   (/)c0 3455   {csn 3640   "cima 4692   -->wf 5251   ` cfv 5255  (class class class)co 5858  TopOnctopon 16632   neicnei 16834   Filcfil 17540    fClusf cfcf 17632
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-int 3863  df-iun 3907  df-iin 3908  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-cld 16756  df-ntr 16757  df-cls 16758  df-nei 16835  df-fbas 17520  df-fg 17521  df-fil 17541  df-fm 17633  df-fcls 17636  df-fcf 17637
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