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Theorem isflf 20619
Description: The property of being a limit point of a function. (Contributed by Jeff Hankins, 8-Nov-2009.) (Revised by Stefan O'Rear, 7-Aug-2015.)
Assertion
Ref Expression
isflf  |-  ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
)  /\  F : Y
--> X )  ->  ( A  e.  ( ( J  fLimf  L ) `  F )  <->  ( A  e.  X  /\  A. o  e.  J  ( A  e.  o  ->  E. s  e.  L  ( F " s )  C_  o
) ) ) )
Distinct variable groups:    A, o    o, s, F    o, J, s    o, L, s    o, X, s    o, Y, s
Allowed substitution hint:    A( s)

Proof of Theorem isflf
StepHypRef Expression
1 flfval 20616 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
)  /\  F : Y
--> X )  ->  (
( J  fLimf  L ) `
 F )  =  ( J  fLim  (
( X  FilMap  F ) `
 L ) ) )
21eleq2d 2527 . 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 19555 . . . . 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 20474 . . . . 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 20570 . . . 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 flimopn 20601 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  (
( X  FilMap  F ) `
 L )  e.  ( Fil `  X
) )  ->  ( A  e.  ( J  fLim  ( ( X  FilMap  F ) `  L ) )  <->  ( A  e.  X  /\  A. o  e.  J  ( A  e.  o  ->  o  e.  ( ( 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  /\  A. o  e.  J  ( A  e.  o  ->  o  e.  ( ( X  FilMap  F ) `  L ) ) ) ) )
13 elfm 20573 . . . . . . . 8  |-  ( ( X  e.  J  /\  L  e.  ( fBas `  Y )  /\  F : Y --> X )  -> 
( o  e.  ( ( X  FilMap  F ) `
 L )  <->  ( o  C_  X  /\  E. s  e.  L  ( F " s )  C_  o
) ) )
145, 7, 8, 13syl3anc 1228 . . . . . . 7  |-  ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
)  /\  F : Y
--> X )  ->  (
o  e.  ( ( X  FilMap  F ) `  L )  <->  ( o  C_  X  /\  E. s  e.  L  ( F " s )  C_  o
) ) )
1514adantr 465 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
)  /\  F : Y
--> X )  /\  o  e.  J )  ->  (
o  e.  ( ( X  FilMap  F ) `  L )  <->  ( o  C_  X  /\  E. s  e.  L  ( F " s )  C_  o
) ) )
16 toponss 19556 . . . . . . . 8  |-  ( ( J  e.  (TopOn `  X )  /\  o  e.  J )  ->  o  C_  X )
173, 16sylan 471 . . . . . . 7  |-  ( ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
)  /\  F : Y
--> X )  /\  o  e.  J )  ->  o  C_  X )
1817biantrurd 508 . . . . . 6  |-  ( ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
)  /\  F : Y
--> X )  /\  o  e.  J )  ->  ( E. s  e.  L  ( F " s ) 
C_  o  <->  ( o  C_  X  /\  E. s  e.  L  ( F " s )  C_  o
) ) )
1915, 18bitr4d 256 . . . . 5  |-  ( ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
)  /\  F : Y
--> X )  /\  o  e.  J )  ->  (
o  e.  ( ( X  FilMap  F ) `  L )  <->  E. s  e.  L  ( F " s )  C_  o
) )
2019imbi2d 316 . . . 4  |-  ( ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
)  /\  F : Y
--> X )  /\  o  e.  J )  ->  (
( A  e.  o  ->  o  e.  ( ( X  FilMap  F ) `
 L ) )  <-> 
( A  e.  o  ->  E. s  e.  L  ( F " s ) 
C_  o ) ) )
2120ralbidva 2893 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
)  /\  F : Y
--> X )  ->  ( A. o  e.  J  ( A  e.  o  ->  o  e.  ( ( X  FilMap  F ) `  L ) )  <->  A. o  e.  J  ( A  e.  o  ->  E. s  e.  L  ( F " s )  C_  o
) ) )
2221anbi2d 703 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
)  /\  F : Y
--> X )  ->  (
( A  e.  X  /\  A. o  e.  J  ( A  e.  o  ->  o  e.  ( ( X  FilMap  F ) `  L ) ) )  <-> 
( A  e.  X  /\  A. o  e.  J  ( A  e.  o  ->  E. s  e.  L  ( F " s ) 
C_  o ) ) ) )
232, 12, 223bitrd 279 1  |-  ( ( J  e.  (TopOn `  X )  /\  L  e.  ( Fil `  Y
)  /\  F : Y
--> X )  ->  ( A  e.  ( ( J  fLimf  L ) `  F )  <->  ( A  e.  X  /\  A. o  e.  J  ( A  e.  o  ->  E. s  e.  L  ( F " s )  C_  o
) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    e. wcel 1819   A.wral 2807   E.wrex 2808    C_ wss 3471   "cima 5011   -->wf 5590   ` cfv 5594  (class class class)co 6296   fBascfbas 18532  TopOnctopon 19521   Filcfil 20471    FilMap cfm 20559    fLim cflim 20560    fLimf cflf 20561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-nel 2655  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  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 6299  df-oprab 6300  df-mpt2 6301  df-map 7440  df-fbas 18542  df-fg 18543  df-top 19525  df-topon 19528  df-ntr 19647  df-nei 19725  df-fil 20472  df-fm 20564  df-flim 20565  df-flf 20566
This theorem is referenced by:  flfelbas  20620  flffbas  20621  flftg  20622  cnpflfi  20625  cnpflf2  20626  txflf  20632  limcflf  22410  rrhre  28152
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