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Theorem flfnei 17702
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 17701 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
)  /\  F : Y
--> X )  ->  (
( J  fLimf  L ) `
 F )  =  ( J  fLim  (
( X  FilMap  F ) `
 L ) ) )
21eleq2d 2363 . 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 955 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
)  /\  F : Y
--> X )  ->  J  e.  (TopOn `  X )
)
4 toponmax 16682 . . . . 5  |-  ( J  e.  (TopOn `  X
)  ->  X  e.  J )
543ad2ant1 976 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
)  /\  F : Y
--> X )  ->  X  e.  J )
6 filfbas 17559 . . . . 5  |-  ( L  e.  ( Fil `  Y
)  ->  L  e.  ( fBas `  Y )
)
763ad2ant2 977 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
)  /\  F : Y
--> X )  ->  L  e.  ( fBas `  Y
) )
8 simp3 957 . . . 4  |-  ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
)  /\  F : Y
--> X )  ->  F : Y --> X )
9 fmfil 17655 . . . 4  |-  ( ( X  e.  J  /\  L  e.  ( fBas `  Y )  /\  F : Y --> X )  -> 
( ( X  FilMap  F ) `  L )  e.  ( Fil `  X
) )
105, 7, 8, 9syl3anc 1182 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
)  /\  F : Y
--> X )  ->  (
( X  FilMap  F ) `
 L )  e.  ( Fil `  X
) )
11 elflim 17682 . . 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 642 . 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 3183 . . . 4  |-  ( ( ( nei `  J
) `  { A } )  C_  (
( X  FilMap  F ) `
 L )  <->  A. n  e.  ( ( nei `  J
) `  { A } ) n  e.  ( ( X  FilMap  F ) `  L ) )
14 topontop 16680 . . . . . . . . 9  |-  ( J  e.  (TopOn `  X
)  ->  J  e.  Top )
15143ad2ant1 976 . . . . . . . 8  |-  ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
)  /\  F : Y
--> X )  ->  J  e.  Top )
16 eqid 2296 . . . . . . . . 9  |-  U. J  =  U. J
1716neii1 16859 . . . . . . . 8  |-  ( ( J  e.  Top  /\  n  e.  ( ( nei `  J ) `  { A } ) )  ->  n  C_  U. J
)
1815, 17sylan 457 . . . . . . 7  |-  ( ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
)  /\  F : Y
--> X )  /\  n  e.  ( ( nei `  J
) `  { A } ) )  ->  n  C_  U. J )
19 toponuni 16681 . . . . . . . . 9  |-  ( J  e.  (TopOn `  X
)  ->  X  =  U. J )
20193ad2ant1 976 . . . . . . . 8  |-  ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
)  /\  F : Y
--> X )  ->  X  =  U. J )
2120adantr 451 . . . . . . 7  |-  ( ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
)  /\  F : Y
--> X )  /\  n  e.  ( ( nei `  J
) `  { A } ) )  ->  X  =  U. J )
2218, 21sseqtr4d 3228 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
)  /\  F : Y
--> X )  /\  n  e.  ( ( nei `  J
) `  { A } ) )  ->  n  C_  X )
23 elfm 17658 . . . . . . . 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 1182 . . . . . . 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 875 . . . . . 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 456 . . . . 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 2572 . . . 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 248 . . 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 684 . 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 270 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 176    /\ wa 358    /\ w3a 934    = wceq 1632    e. wcel 1696   A.wral 2556   E.wrex 2557    C_ wss 3165   {csn 3653   U.cuni 3843   "cima 4708   -->wf 5267   ` cfv 5271  (class class class)co 5874   Topctop 16647  TopOnctopon 16648   neicnei 16850   fBascfbas 17534   Filcfil 17556    FilMap cfm 17644    fLim cflim 17645    fLimf cflf 17646
This theorem is referenced by:  flfneii  17703  limptlimpr2lem1  25677  limptlimpr2lem2  25678  flfnei2  25680
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-rep 4147  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-iun 3923  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-map 6790  df-top 16652  df-topon 16655  df-nei 16851  df-fbas 17536  df-fg 17537  df-fil 17557  df-fm 17649  df-flim 17650  df-flf 17651
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