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Theorem flfnei 20360
Description: The property of being a limit point of a function in terms of neighborhoods. (Contributed by Jeff Hankins, 9-Nov-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.)
Assertion
Ref Expression
flfnei  |-  ( ( 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
) ) )
Distinct variable groups:    n, s, F    A, n    n, J, s    n, L, s   
n, X, s    n, Y, s
Allowed substitution hint:    A( s)

Proof of Theorem flfnei
StepHypRef Expression
1 flfval 20359 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
)  /\  F : Y
--> X )  ->  (
( J  fLimf  L ) `
 F )  =  ( J  fLim  (
( X  FilMap  F ) `
 L ) ) )
21eleq2d 2537 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
)  /\  F : Y
--> X )  ->  ( A  e.  ( ( J  fLimf  L ) `  F )  <->  A  e.  ( J  fLim  ( ( X  FilMap  F ) `  L ) ) ) )
3 simp1 996 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
)  /\  F : Y
--> X )  ->  J  e.  (TopOn `  X )
)
4 toponmax 19298 . . . . 5  |-  ( J  e.  (TopOn `  X
)  ->  X  e.  J )
543ad2ant1 1017 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
)  /\  F : Y
--> X )  ->  X  e.  J )
6 filfbas 20217 . . . . 5  |-  ( L  e.  ( Fil `  Y
)  ->  L  e.  ( fBas `  Y )
)
763ad2ant2 1018 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
)  /\  F : Y
--> X )  ->  L  e.  ( fBas `  Y
) )
8 simp3 998 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
)  /\  F : Y
--> X )  ->  F : Y --> X )
9 fmfil 20313 . . . 4  |-  ( ( X  e.  J  /\  L  e.  ( fBas `  Y )  /\  F : Y --> X )  -> 
( ( X  FilMap  F ) `  L )  e.  ( Fil `  X
) )
105, 7, 8, 9syl3anc 1228 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
)  /\  F : Y
--> X )  ->  (
( X  FilMap  F ) `
 L )  e.  ( Fil `  X
) )
11 elflim 20340 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  (
( X  FilMap  F ) `
 L )  e.  ( Fil `  X
) )  ->  ( A  e.  ( J  fLim  ( ( X  FilMap  F ) `  L ) )  <->  ( A  e.  X  /\  ( ( nei `  J ) `
 { A }
)  C_  ( ( X  FilMap  F ) `  L ) ) ) )
123, 10, 11syl2anc 661 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
)  /\  F : Y
--> X )  ->  ( A  e.  ( J  fLim  ( ( X  FilMap  F ) `  L ) )  <->  ( A  e.  X  /\  ( ( nei `  J ) `
 { A }
)  C_  ( ( X  FilMap  F ) `  L ) ) ) )
13 dfss3 3499 . . . 4  |-  ( ( ( nei `  J
) `  { A } )  C_  (
( X  FilMap  F ) `
 L )  <->  A. n  e.  ( ( nei `  J
) `  { A } ) n  e.  ( ( X  FilMap  F ) `  L ) )
14 topontop 19296 . . . . . . . . 9  |-  ( J  e.  (TopOn `  X
)  ->  J  e.  Top )
15143ad2ant1 1017 . . . . . . . 8  |-  ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
)  /\  F : Y
--> X )  ->  J  e.  Top )
16 eqid 2467 . . . . . . . . 9  |-  U. J  =  U. J
1716neii1 19475 . . . . . . . 8  |-  ( ( J  e.  Top  /\  n  e.  ( ( nei `  J ) `  { A } ) )  ->  n  C_  U. J
)
1815, 17sylan 471 . . . . . . 7  |-  ( ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
)  /\  F : Y
--> X )  /\  n  e.  ( ( nei `  J
) `  { A } ) )  ->  n  C_  U. J )
19 toponuni 19297 . . . . . . . . 9  |-  ( J  e.  (TopOn `  X
)  ->  X  =  U. J )
20193ad2ant1 1017 . . . . . . . 8  |-  ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
)  /\  F : Y
--> X )  ->  X  =  U. J )
2120adantr 465 . . . . . . 7  |-  ( ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
)  /\  F : Y
--> X )  /\  n  e.  ( ( nei `  J
) `  { A } ) )  ->  X  =  U. J )
2218, 21sseqtr4d 3546 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
)  /\  F : Y
--> X )  /\  n  e.  ( ( nei `  J
) `  { A } ) )  ->  n  C_  X )
23 elfm 20316 . . . . . . . 8  |-  ( ( X  e.  J  /\  L  e.  ( fBas `  Y )  /\  F : Y --> X )  -> 
( n  e.  ( ( X  FilMap  F ) `
 L )  <->  ( n  C_  X  /\  E. s  e.  L  ( F " s )  C_  n
) ) )
245, 7, 8, 23syl3anc 1228 . . . . . . 7  |-  ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
)  /\  F : Y
--> X )  ->  (
n  e.  ( ( X  FilMap  F ) `  L )  <->  ( n  C_  X  /\  E. s  e.  L  ( F " s )  C_  n
) ) )
2524baibd 907 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
)  /\  F : Y
--> X )  /\  n  C_  X )  ->  (
n  e.  ( ( X  FilMap  F ) `  L )  <->  E. s  e.  L  ( F " s )  C_  n
) )
2622, 25syldan 470 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
)  /\  F : Y
--> X )  /\  n  e.  ( ( nei `  J
) `  { A } ) )  -> 
( n  e.  ( ( X  FilMap  F ) `
 L )  <->  E. s  e.  L  ( F " s )  C_  n
) )
2726ralbidva 2903 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
)  /\  F : Y
--> X )  ->  ( A. n  e.  (
( nei `  J
) `  { A } ) n  e.  ( ( X  FilMap  F ) `  L )  <->  A. n  e.  (
( nei `  J
) `  { A } ) E. s  e.  L  ( F " s )  C_  n
) )
2813, 27syl5bb 257 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
)  /\  F : Y
--> X )  ->  (
( ( nei `  J
) `  { A } )  C_  (
( X  FilMap  F ) `
 L )  <->  A. n  e.  ( ( nei `  J
) `  { A } ) E. s  e.  L  ( F " s )  C_  n
) )
2928anbi2d 703 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
)  /\  F : Y
--> X )  ->  (
( A  e.  X  /\  ( ( nei `  J
) `  { A } )  C_  (
( X  FilMap  F ) `
 L ) )  <-> 
( A  e.  X  /\  A. n  e.  ( ( nei `  J
) `  { A } ) E. s  e.  L  ( F " s )  C_  n
) ) )
302, 12, 293bitrd 279 1  |-  ( ( 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
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   A.wral 2817   E.wrex 2818    C_ wss 3481   {csn 4033   U.cuni 4251   "cima 5008   -->wf 5590   ` cfv 5594  (class class class)co 6295   fBascfbas 18276   Topctop 19263  TopOnctopon 19264   neicnei 19466   Filcfil 20214    FilMap cfm 20302    fLim cflim 20303    fLimf cflf 20304
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 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
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 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-map 7434  df-fbas 18286  df-fg 18287  df-top 19268  df-topon 19271  df-nei 19467  df-fil 20215  df-fm 20307  df-flim 20308  df-flf 20309
This theorem is referenced by:  flfneii  20361  cnextcn  20435  cnextfres  20436
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