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Theorem fcfneii 18069
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 18067 . . 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 3535 . . . . . . . 8  |-  ( n  =  N  ->  (
n  i^i  ( F " s ) )  =  ( N  i^i  ( F " s ) ) )
32neeq1d 2614 . . . . . . 7  |-  ( n  =  N  ->  (
( n  i^i  ( F " s ) )  =/=  (/)  <->  ( N  i^i  ( F " s ) )  =/=  (/) ) )
4 imaeq2 5199 . . . . . . . . 9  |-  ( s  =  S  ->  ( F " s )  =  ( F " S
) )
54ineq2d 3542 . . . . . . . 8  |-  ( s  =  S  ->  ( N  i^i  ( F "
s ) )  =  ( N  i^i  ( F " S ) ) )
65neeq1d 2614 . . . . . . 7  |-  ( s  =  S  ->  (
( N  i^i  ( F " s ) )  =/=  (/)  <->  ( N  i^i  ( F " S ) )  =/=  (/) ) )
73, 6rspc2v 3058 . . . . . 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 424 . . . . 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 75 . . . 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 453 . . 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 220 . 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 1168 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 359    /\ w3a 936    = wceq 1652    e. wcel 1725    =/= wne 2599   A.wral 2705    i^i cin 3319   (/)c0 3628   {csn 3814   "cima 4881   -->wf 5450   ` cfv 5454  (class class class)co 6081  TopOnctopon 16959   neicnei 17161   Filcfil 17877    fClusf cfcf 17969
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-rep 4320  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-nel 2602  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-int 4051  df-iun 4095  df-iin 4096  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-map 7020  df-fbas 16699  df-fg 16700  df-top 16963  df-topon 16966  df-cld 17083  df-ntr 17084  df-cls 17085  df-nei 17162  df-fil 17878  df-fm 17970  df-fcls 17973  df-fcf 17974
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