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Theorem flfnei 18025
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 18024 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
)  /\  F : Y
--> X )  ->  (
( J  fLimf  L ) `
 F )  =  ( J  fLim  (
( X  FilMap  F ) `
 L ) ) )
21eleq2d 2505 . 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 958 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
)  /\  F : Y
--> X )  ->  J  e.  (TopOn `  X )
)
4 toponmax 16995 . . . . 5  |-  ( J  e.  (TopOn `  X
)  ->  X  e.  J )
543ad2ant1 979 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
)  /\  F : Y
--> X )  ->  X  e.  J )
6 filfbas 17882 . . . . 5  |-  ( L  e.  ( Fil `  Y
)  ->  L  e.  ( fBas `  Y )
)
763ad2ant2 980 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
)  /\  F : Y
--> X )  ->  L  e.  ( fBas `  Y
) )
8 simp3 960 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
)  /\  F : Y
--> X )  ->  F : Y --> X )
9 fmfil 17978 . . . 4  |-  ( ( X  e.  J  /\  L  e.  ( fBas `  Y )  /\  F : Y --> X )  -> 
( ( X  FilMap  F ) `  L )  e.  ( Fil `  X
) )
105, 7, 8, 9syl3anc 1185 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
)  /\  F : Y
--> X )  ->  (
( X  FilMap  F ) `
 L )  e.  ( Fil `  X
) )
11 elflim 18005 . . 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 644 . 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 3340 . . . 4  |-  ( ( ( nei `  J
) `  { A } )  C_  (
( X  FilMap  F ) `
 L )  <->  A. n  e.  ( ( nei `  J
) `  { A } ) n  e.  ( ( X  FilMap  F ) `  L ) )
14 topontop 16993 . . . . . . . . 9  |-  ( J  e.  (TopOn `  X
)  ->  J  e.  Top )
15143ad2ant1 979 . . . . . . . 8  |-  ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
)  /\  F : Y
--> X )  ->  J  e.  Top )
16 eqid 2438 . . . . . . . . 9  |-  U. J  =  U. J
1716neii1 17172 . . . . . . . 8  |-  ( ( J  e.  Top  /\  n  e.  ( ( nei `  J ) `  { A } ) )  ->  n  C_  U. J
)
1815, 17sylan 459 . . . . . . 7  |-  ( ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
)  /\  F : Y
--> X )  /\  n  e.  ( ( nei `  J
) `  { A } ) )  ->  n  C_  U. J )
19 toponuni 16994 . . . . . . . . 9  |-  ( J  e.  (TopOn `  X
)  ->  X  =  U. J )
20193ad2ant1 979 . . . . . . . 8  |-  ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
)  /\  F : Y
--> X )  ->  X  =  U. J )
2120adantr 453 . . . . . . 7  |-  ( ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
)  /\  F : Y
--> X )  /\  n  e.  ( ( nei `  J
) `  { A } ) )  ->  X  =  U. J )
2218, 21sseqtr4d 3387 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
)  /\  F : Y
--> X )  /\  n  e.  ( ( nei `  J
) `  { A } ) )  ->  n  C_  X )
23 elfm 17981 . . . . . . . 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 1185 . . . . . . 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 877 . . . . . 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 458 . . . . 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 2723 . . . 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 250 . . 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 686 . 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 272 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 set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    = wceq 1653    e. wcel 1726   A.wral 2707   E.wrex 2708    C_ wss 3322   {csn 3816   U.cuni 4017   "cima 4883   -->wf 5452   ` cfv 5456  (class class class)co 6083   fBascfbas 16691   Topctop 16960  TopOnctopon 16961   neicnei 17163   Filcfil 17879    FilMap cfm 17967    fLim cflim 17968    fLimf cflf 17969
This theorem is referenced by:  flfneii  18026  cnextcn  18100  cnextfres  18101
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4322  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405  ax-un 4703
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-rn 4891  df-res 4892  df-ima 4893  df-iota 5420  df-fun 5458  df-fn 5459  df-f 5460  df-f1 5461  df-fo 5462  df-f1o 5463  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-map 7022  df-fbas 16701  df-fg 16702  df-top 16965  df-topon 16968  df-nei 17164  df-fil 17880  df-fm 17972  df-flim 17973  df-flf 17974
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