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Theorem flfnei2 25680
Description: The property of being a limit point of a function in terms of filter and of preimage of a neighborhood. (Contributed by FL, 13-Dec-2013.) (Revised by Mario Carneiro, 8-Aug-2015.)
Hypothesis
Ref Expression
flfnei2.x  |-  X  = 
U. J
Assertion
Ref Expression
flfnei2  |-  ( ( J  e.  Top  /\  L  e.  ( Fil `  Y )  /\  F : Y --> X )  -> 
( A  e.  ( ( J  fLimf  L ) `
 F )  <->  ( A  e.  X  /\  A. n  e.  ( ( nei `  J
) `  { A } ) ( `' F " n )  e.  L ) ) )
Distinct variable groups:    A, n    n, F    n, J    n, L    n, X    n, Y

Proof of Theorem flfnei2
Dummy variable  l is distinct from all other variables.
StepHypRef Expression
1 flfnei2.x . . . 4  |-  X  = 
U. J
21toptopon 16687 . . 3  |-  ( J  e.  Top  <->  J  e.  (TopOn `  X ) )
3 flfnei 17702 . . 3  |-  ( ( 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. l  e.  L  ( F " l )  C_  n
) ) )
42, 3syl3an1b 1218 . 2  |-  ( ( J  e.  Top  /\  L  e.  ( Fil `  Y )  /\  F : Y --> X )  -> 
( A  e.  ( ( J  fLimf  L ) `
 F )  <->  ( A  e.  X  /\  A. n  e.  ( ( nei `  J
) `  { A } ) E. l  e.  L  ( F " l )  C_  n
) ) )
5 simpl2 959 . . . . . . . 8  |-  ( ( ( J  e.  Top  /\  L  e.  ( Fil `  Y )  /\  F : Y --> X )  /\  ( l  e.  L  /\  ( F " l
)  C_  n )
)  ->  L  e.  ( Fil `  Y ) )
6 simprl 732 . . . . . . . 8  |-  ( ( ( J  e.  Top  /\  L  e.  ( Fil `  Y )  /\  F : Y --> X )  /\  ( l  e.  L  /\  ( F " l
)  C_  n )
)  ->  l  e.  L )
7 cnvimass 5049 . . . . . . . . 9  |-  ( `' F " n ) 
C_  dom  F
8 simpl3 960 . . . . . . . . . 10  |-  ( ( ( J  e.  Top  /\  L  e.  ( Fil `  Y )  /\  F : Y --> X )  /\  ( l  e.  L  /\  ( F " l
)  C_  n )
)  ->  F : Y
--> X )
9 fdm 5409 . . . . . . . . . 10  |-  ( F : Y --> X  ->  dom  F  =  Y )
108, 9syl 15 . . . . . . . . 9  |-  ( ( ( J  e.  Top  /\  L  e.  ( Fil `  Y )  /\  F : Y --> X )  /\  ( l  e.  L  /\  ( F " l
)  C_  n )
)  ->  dom  F  =  Y )
117, 10syl5sseq 3239 . . . . . . . 8  |-  ( ( ( J  e.  Top  /\  L  e.  ( Fil `  Y )  /\  F : Y --> X )  /\  ( l  e.  L  /\  ( F " l
)  C_  n )
)  ->  ( `' F " n )  C_  Y )
12 simpl3 960 . . . . . . . . . . . 12  |-  ( ( ( J  e.  Top  /\  L  e.  ( Fil `  Y )  /\  F : Y --> X )  /\  l  e.  L )  ->  F : Y --> X )
13 ffun 5407 . . . . . . . . . . . 12  |-  ( F : Y --> X  ->  Fun  F )
1412, 13syl 15 . . . . . . . . . . 11  |-  ( ( ( J  e.  Top  /\  L  e.  ( Fil `  Y )  /\  F : Y --> X )  /\  l  e.  L )  ->  Fun  F )
15 filelss 17563 . . . . . . . . . . . . 13  |-  ( ( L  e.  ( Fil `  Y )  /\  l  e.  L )  ->  l  C_  Y )
16153ad2antl2 1118 . . . . . . . . . . . 12  |-  ( ( ( J  e.  Top  /\  L  e.  ( Fil `  Y )  /\  F : Y --> X )  /\  l  e.  L )  ->  l  C_  Y )
1712, 9syl 15 . . . . . . . . . . . 12  |-  ( ( ( J  e.  Top  /\  L  e.  ( Fil `  Y )  /\  F : Y --> X )  /\  l  e.  L )  ->  dom  F  =  Y )
1816, 17sseqtr4d 3228 . . . . . . . . . . 11  |-  ( ( ( J  e.  Top  /\  L  e.  ( Fil `  Y )  /\  F : Y --> X )  /\  l  e.  L )  ->  l  C_  dom  F )
19 funimass3 5657 . . . . . . . . . . 11  |-  ( ( Fun  F  /\  l  C_ 
dom  F )  -> 
( ( F "
l )  C_  n  <->  l 
C_  ( `' F " n ) ) )
2014, 18, 19syl2anc 642 . . . . . . . . . 10  |-  ( ( ( J  e.  Top  /\  L  e.  ( Fil `  Y )  /\  F : Y --> X )  /\  l  e.  L )  ->  ( ( F "
l )  C_  n  <->  l 
C_  ( `' F " n ) ) )
2120biimpd 198 . . . . . . . . 9  |-  ( ( ( J  e.  Top  /\  L  e.  ( Fil `  Y )  /\  F : Y --> X )  /\  l  e.  L )  ->  ( ( F "
l )  C_  n  ->  l  C_  ( `' F " n ) ) )
2221impr 602 . . . . . . . 8  |-  ( ( ( J  e.  Top  /\  L  e.  ( Fil `  Y )  /\  F : Y --> X )  /\  ( l  e.  L  /\  ( F " l
)  C_  n )
)  ->  l  C_  ( `' F " n ) )
23 filss 17564 . . . . . . . 8  |-  ( ( L  e.  ( Fil `  Y )  /\  (
l  e.  L  /\  ( `' F " n ) 
C_  Y  /\  l  C_  ( `' F "
n ) ) )  ->  ( `' F " n )  e.  L
)
245, 6, 11, 22, 23syl13anc 1184 . . . . . . 7  |-  ( ( ( J  e.  Top  /\  L  e.  ( Fil `  Y )  /\  F : Y --> X )  /\  ( l  e.  L  /\  ( F " l
)  C_  n )
)  ->  ( `' F " n )  e.  L )
2524expr 598 . . . . . 6  |-  ( ( ( J  e.  Top  /\  L  e.  ( Fil `  Y )  /\  F : Y --> X )  /\  l  e.  L )  ->  ( ( F "
l )  C_  n  ->  ( `' F "
n )  e.  L
) )
2625rexlimdva 2680 . . . . 5  |-  ( ( J  e.  Top  /\  L  e.  ( Fil `  Y )  /\  F : Y --> X )  -> 
( E. l  e.  L  ( F "
l )  C_  n  ->  ( `' F "
n )  e.  L
) )
27133ad2ant3 978 . . . . . . 7  |-  ( ( J  e.  Top  /\  L  e.  ( Fil `  Y )  /\  F : Y --> X )  ->  Fun  F )
28 funimacnv 5340 . . . . . . . 8  |-  ( Fun 
F  ->  ( F " ( `' F "
n ) )  =  ( n  i^i  ran  F ) )
29 inss1 3402 . . . . . . . . 9  |-  ( n  i^i  ran  F )  C_  n
3029a1i 10 . . . . . . . 8  |-  ( Fun 
F  ->  ( n  i^i  ran  F )  C_  n )
3128, 30eqsstrd 3225 . . . . . . 7  |-  ( Fun 
F  ->  ( F " ( `' F "
n ) )  C_  n )
3227, 31syl 15 . . . . . 6  |-  ( ( J  e.  Top  /\  L  e.  ( Fil `  Y )  /\  F : Y --> X )  -> 
( F " ( `' F " n ) )  C_  n )
33 imaeq2 5024 . . . . . . . . 9  |-  ( l  =  ( `' F " n )  ->  ( F " l )  =  ( F " ( `' F " n ) ) )
3433sseq1d 3218 . . . . . . . 8  |-  ( l  =  ( `' F " n )  ->  (
( F " l
)  C_  n  <->  ( F " ( `' F "
n ) )  C_  n ) )
3534rspcev 2897 . . . . . . 7  |-  ( ( ( `' F "
n )  e.  L  /\  ( F " ( `' F " n ) )  C_  n )  ->  E. l  e.  L  ( F " l ) 
C_  n )
3635expcom 424 . . . . . 6  |-  ( ( F " ( `' F " n ) )  C_  n  ->  ( ( `' F "
n )  e.  L  ->  E. l  e.  L  ( F " l ) 
C_  n ) )
3732, 36syl 15 . . . . 5  |-  ( ( J  e.  Top  /\  L  e.  ( Fil `  Y )  /\  F : Y --> X )  -> 
( ( `' F " n )  e.  L  ->  E. l  e.  L  ( F " l ) 
C_  n ) )
3826, 37impbid 183 . . . 4  |-  ( ( J  e.  Top  /\  L  e.  ( Fil `  Y )  /\  F : Y --> X )  -> 
( E. l  e.  L  ( F "
l )  C_  n  <->  ( `' F " n )  e.  L ) )
3938ralbidv 2576 . . 3  |-  ( ( J  e.  Top  /\  L  e.  ( Fil `  Y )  /\  F : Y --> X )  -> 
( A. n  e.  ( ( nei `  J
) `  { A } ) E. l  e.  L  ( F " l )  C_  n  <->  A. n  e.  ( ( nei `  J ) `
 { A }
) ( `' F " n )  e.  L
) )
4039anbi2d 684 . 2  |-  ( ( J  e.  Top  /\  L  e.  ( Fil `  Y )  /\  F : Y --> X )  -> 
( ( A  e.  X  /\  A. n  e.  ( ( nei `  J
) `  { A } ) E. l  e.  L  ( F " l )  C_  n
)  <->  ( A  e.  X  /\  A. n  e.  ( ( nei `  J
) `  { A } ) ( `' F " n )  e.  L ) ) )
414, 40bitrd 244 1  |-  ( ( J  e.  Top  /\  L  e.  ( Fil `  Y )  /\  F : Y --> X )  -> 
( A  e.  ( ( J  fLimf  L ) `
 F )  <->  ( A  e.  X  /\  A. n  e.  ( ( nei `  J
) `  { A } ) ( `' F " n )  e.  L ) ) )
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    i^i cin 3164    C_ wss 3165   {csn 3653   U.cuni 3843   `'ccnv 4704   dom cdm 4705   ran crn 4706   "cima 4708   Fun wfun 5265   -->wf 5267   ` cfv 5271  (class class class)co 5874   Topctop 16647  TopOnctopon 16648   neicnei 16850   Filcfil 17556    fLimf cflf 17646
This theorem is referenced by:  islimrs3  25684  islimrs4  25685
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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