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Theorem flfnei 19406
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 19405 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
)  /\  F : Y
--> X )  ->  (
( J  fLimf  L ) `
 F )  =  ( J  fLim  (
( X  FilMap  F ) `
 L ) ) )
21eleq2d 2500 . 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 981 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
)  /\  F : Y
--> X )  ->  J  e.  (TopOn `  X )
)
4 toponmax 18375 . . . . 5  |-  ( J  e.  (TopOn `  X
)  ->  X  e.  J )
543ad2ant1 1002 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
)  /\  F : Y
--> X )  ->  X  e.  J )
6 filfbas 19263 . . . . 5  |-  ( L  e.  ( Fil `  Y
)  ->  L  e.  ( fBas `  Y )
)
763ad2ant2 1003 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
)  /\  F : Y
--> X )  ->  L  e.  ( fBas `  Y
) )
8 simp3 983 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
)  /\  F : Y
--> X )  ->  F : Y --> X )
9 fmfil 19359 . . . 4  |-  ( ( X  e.  J  /\  L  e.  ( fBas `  Y )  /\  F : Y --> X )  -> 
( ( X  FilMap  F ) `  L )  e.  ( Fil `  X
) )
105, 7, 8, 9syl3anc 1211 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
)  /\  F : Y
--> X )  ->  (
( X  FilMap  F ) `
 L )  e.  ( Fil `  X
) )
11 elflim 19386 . . 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 654 . 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 3334 . . . 4  |-  ( ( ( nei `  J
) `  { A } )  C_  (
( X  FilMap  F ) `
 L )  <->  A. n  e.  ( ( nei `  J
) `  { A } ) n  e.  ( ( X  FilMap  F ) `  L ) )
14 topontop 18373 . . . . . . . . 9  |-  ( J  e.  (TopOn `  X
)  ->  J  e.  Top )
15143ad2ant1 1002 . . . . . . . 8  |-  ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
)  /\  F : Y
--> X )  ->  J  e.  Top )
16 eqid 2433 . . . . . . . . 9  |-  U. J  =  U. J
1716neii1 18552 . . . . . . . 8  |-  ( ( J  e.  Top  /\  n  e.  ( ( nei `  J ) `  { A } ) )  ->  n  C_  U. J
)
1815, 17sylan 468 . . . . . . 7  |-  ( ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
)  /\  F : Y
--> X )  /\  n  e.  ( ( nei `  J
) `  { A } ) )  ->  n  C_  U. J )
19 toponuni 18374 . . . . . . . . 9  |-  ( J  e.  (TopOn `  X
)  ->  X  =  U. J )
20193ad2ant1 1002 . . . . . . . 8  |-  ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
)  /\  F : Y
--> X )  ->  X  =  U. J )
2120adantr 462 . . . . . . 7  |-  ( ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
)  /\  F : Y
--> X )  /\  n  e.  ( ( nei `  J
) `  { A } ) )  ->  X  =  U. J )
2218, 21sseqtr4d 3381 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
)  /\  F : Y
--> X )  /\  n  e.  ( ( nei `  J
) `  { A } ) )  ->  n  C_  X )
23 elfm 19362 . . . . . . . 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 1211 . . . . . . 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 893 . . . . . 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 467 . . . . 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 2721 . . . 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 696 . 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 958    = wceq 1362    e. wcel 1755   A.wral 2705   E.wrex 2706    C_ wss 3316   {csn 3865   U.cuni 4079   "cima 4830   -->wf 5402   ` cfv 5406  (class class class)co 6080   fBascfbas 17648   Topctop 18340  TopOnctopon 18341   neicnei 18543   Filcfil 19260    FilMap cfm 19348    fLim cflim 19349    fLimf cflf 19350
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1594  ax-4 1605  ax-5 1669  ax-6 1707  ax-7 1727  ax-8 1757  ax-9 1759  ax-10 1774  ax-11 1779  ax-12 1791  ax-13 1942  ax-ext 2414  ax-rep 4391  ax-sep 4401  ax-nul 4409  ax-pow 4458  ax-pr 4519  ax-un 6361
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 960  df-tru 1365  df-ex 1590  df-nf 1593  df-sb 1700  df-eu 2258  df-mo 2259  df-clab 2420  df-cleq 2426  df-clel 2429  df-nfc 2558  df-ne 2598  df-nel 2599  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2964  df-sbc 3176  df-csb 3277  df-dif 3319  df-un 3321  df-in 3323  df-ss 3330  df-nul 3626  df-if 3780  df-pw 3850  df-sn 3866  df-pr 3868  df-op 3872  df-uni 4080  df-iun 4161  df-br 4281  df-opab 4339  df-mpt 4340  df-id 4623  df-xp 4833  df-rel 4834  df-cnv 4835  df-co 4836  df-dm 4837  df-rn 4838  df-res 4839  df-ima 4840  df-iota 5369  df-fun 5408  df-fn 5409  df-f 5410  df-f1 5411  df-fo 5412  df-f1o 5413  df-fv 5414  df-ov 6083  df-oprab 6084  df-mpt2 6085  df-map 7204  df-fbas 17658  df-fg 17659  df-top 18345  df-topon 18348  df-nei 18544  df-fil 19261  df-fm 19353  df-flim 19354  df-flf 19355
This theorem is referenced by:  flfneii  19407  cnextcn  19481  cnextfres  19482
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