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Theorem fcfneii 17784
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 17782 . . 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 3397 . . . . . . . 8  |-  ( n  =  N  ->  (
n  i^i  ( F " s ) )  =  ( N  i^i  ( F " s ) ) )
32neeq1d 2492 . . . . . . 7  |-  ( n  =  N  ->  (
( n  i^i  ( F " s ) )  =/=  (/)  <->  ( N  i^i  ( F " s ) )  =/=  (/) ) )
4 imaeq2 5045 . . . . . . . . 9  |-  ( s  =  S  ->  ( F " s )  =  ( F " S
) )
54ineq2d 3404 . . . . . . . 8  |-  ( s  =  S  ->  ( N  i^i  ( F "
s ) )  =  ( N  i^i  ( F " S ) ) )
65neeq1d 2492 . . . . . . 7  |-  ( s  =  S  ->  (
( N  i^i  ( F " s ) )  =/=  (/)  <->  ( N  i^i  ( F " S ) )  =/=  (/) ) )
73, 6rspc2v 2924 . . . . . 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 1633    e. wcel 1701    =/= wne 2479   A.wral 2577    i^i cin 3185   (/)c0 3489   {csn 3674   "cima 4729   -->wf 5288   ` cfv 5292  (class class class)co 5900  TopOnctopon 16688   neicnei 16890   Filcfil 17592    fClusf cfcf 17684
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-rep 4168  ax-sep 4178  ax-nul 4186  ax-pow 4225  ax-pr 4251  ax-un 4549
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-nel 2482  df-ral 2582  df-rex 2583  df-reu 2584  df-rab 2586  df-v 2824  df-sbc 3026  df-csb 3116  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-int 3900  df-iun 3944  df-iin 3945  df-br 4061  df-opab 4115  df-mpt 4116  df-id 4346  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-ov 5903  df-oprab 5904  df-mpt2 5905  df-map 6817  df-fbas 16429  df-fg 16430  df-top 16692  df-topon 16695  df-cld 16812  df-ntr 16813  df-cls 16814  df-nei 16891  df-fil 17593  df-fm 17685  df-fcls 17688  df-fcf 17689
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