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Theorem fbflim 17884
Description: A condition for a filter to converge to a point involving one of its bases. (Contributed by Jeff Hankins, 4-Sep-2009.) (Revised by Stefan O'Rear, 6-Aug-2015.)
Hypothesis
Ref Expression
fbflim.3  |-  F  =  ( X filGen B )
Assertion
Ref Expression
fbflim  |-  ( ( J  e.  (TopOn `  X )  /\  B  e.  ( fBas `  X
) )  ->  ( A  e.  ( J  fLim  F )  <->  ( A  e.  X  /\  A. x  e.  J  ( A  e.  x  ->  E. y  e.  B  y  C_  x ) ) ) )
Distinct variable groups:    x, y, A    x, B, y    x, J, y    x, X, y   
x, F, y

Proof of Theorem fbflim
StepHypRef Expression
1 fbflim.3 . . . 4  |-  F  =  ( X filGen B )
2 fgcl 17786 . . . 4  |-  ( B  e.  ( fBas `  X
)  ->  ( X filGen B )  e.  ( Fil `  X ) )
31, 2syl5eqel 2450 . . 3  |-  ( B  e.  ( fBas `  X
)  ->  F  e.  ( Fil `  X ) )
4 flimopn 17883 . . 3  |-  ( ( J  e.  (TopOn `  X )  /\  F  e.  ( Fil `  X
) )  ->  ( A  e.  ( J  fLim  F )  <->  ( A  e.  X  /\  A. x  e.  J  ( A  e.  x  ->  x  e.  F ) ) ) )
53, 4sylan2 460 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  B  e.  ( fBas `  X
) )  ->  ( A  e.  ( J  fLim  F )  <->  ( A  e.  X  /\  A. x  e.  J  ( A  e.  x  ->  x  e.  F ) ) ) )
61eleq2i 2430 . . . . . . 7  |-  ( x  e.  F  <->  x  e.  ( X filGen B ) )
7 simpllr 735 . . . . . . . 8  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  B  e.  ( fBas `  X ) )  /\  A  e.  X
)  /\  x  e.  J )  ->  B  e.  ( fBas `  X
) )
8 elfg 17779 . . . . . . . 8  |-  ( B  e.  ( fBas `  X
)  ->  ( x  e.  ( X filGen B )  <-> 
( x  C_  X  /\  E. y  e.  B  y  C_  x ) ) )
97, 8syl 15 . . . . . . 7  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  B  e.  ( fBas `  X ) )  /\  A  e.  X
)  /\  x  e.  J )  ->  (
x  e.  ( X
filGen B )  <->  ( x  C_  X  /\  E. y  e.  B  y  C_  x ) ) )
106, 9syl5bb 248 . . . . . 6  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  B  e.  ( fBas `  X ) )  /\  A  e.  X
)  /\  x  e.  J )  ->  (
x  e.  F  <->  ( x  C_  X  /\  E. y  e.  B  y  C_  x ) ) )
11 simpll 730 . . . . . . . 8  |-  ( ( ( J  e.  (TopOn `  X )  /\  B  e.  ( fBas `  X
) )  /\  A  e.  X )  ->  J  e.  (TopOn `  X )
)
12 toponss 16884 . . . . . . . 8  |-  ( ( J  e.  (TopOn `  X )  /\  x  e.  J )  ->  x  C_  X )
1311, 12sylan 457 . . . . . . 7  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  B  e.  ( fBas `  X ) )  /\  A  e.  X
)  /\  x  e.  J )  ->  x  C_  X )
1413biantrurd 494 . . . . . 6  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  B  e.  ( fBas `  X ) )  /\  A  e.  X
)  /\  x  e.  J )  ->  ( E. y  e.  B  y  C_  x  <->  ( x  C_  X  /\  E. y  e.  B  y  C_  x ) ) )
1510, 14bitr4d 247 . . . . 5  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  B  e.  ( fBas `  X ) )  /\  A  e.  X
)  /\  x  e.  J )  ->  (
x  e.  F  <->  E. y  e.  B  y  C_  x ) )
1615imbi2d 307 . . . 4  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  B  e.  ( fBas `  X ) )  /\  A  e.  X
)  /\  x  e.  J )  ->  (
( A  e.  x  ->  x  e.  F )  <-> 
( A  e.  x  ->  E. y  e.  B  y  C_  x ) ) )
1716ralbidva 2644 . . 3  |-  ( ( ( J  e.  (TopOn `  X )  /\  B  e.  ( fBas `  X
) )  /\  A  e.  X )  ->  ( A. x  e.  J  ( A  e.  x  ->  x  e.  F )  <->  A. x  e.  J  ( A  e.  x  ->  E. y  e.  B  y  C_  x ) ) )
1817pm5.32da 622 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  B  e.  ( fBas `  X
) )  ->  (
( A  e.  X  /\  A. x  e.  J  ( A  e.  x  ->  x  e.  F ) )  <->  ( A  e.  X  /\  A. x  e.  J  ( A  e.  x  ->  E. y  e.  B  y  C_  x ) ) ) )
195, 18bitrd 244 1  |-  ( ( J  e.  (TopOn `  X )  /\  B  e.  ( fBas `  X
) )  ->  ( A  e.  ( J  fLim  F )  <->  ( A  e.  X  /\  A. x  e.  J  ( A  e.  x  ->  E. y  e.  B  y  C_  x ) ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1647    e. wcel 1715   A.wral 2628   E.wrex 2629    C_ wss 3238   ` cfv 5358  (class class class)co 5981   fBascfbas 16582   filGencfg 16583  TopOnctopon 16849   Filcfil 17753    fLim cflim 17842
This theorem is referenced by:  fbflim2  17885
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-13 1717  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-rep 4233  ax-sep 4243  ax-nul 4251  ax-pow 4290  ax-pr 4316  ax-un 4615
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-nel 2532  df-ral 2633  df-rex 2634  df-reu 2635  df-rab 2637  df-v 2875  df-sbc 3078  df-csb 3168  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-nul 3544  df-if 3655  df-pw 3716  df-sn 3735  df-pr 3736  df-op 3738  df-uni 3930  df-iun 4009  df-br 4126  df-opab 4180  df-mpt 4181  df-id 4412  df-xp 4798  df-rel 4799  df-cnv 4800  df-co 4801  df-dm 4802  df-rn 4803  df-res 4804  df-ima 4805  df-iota 5322  df-fun 5360  df-fn 5361  df-f 5362  df-f1 5363  df-fo 5364  df-f1o 5365  df-fv 5366  df-ov 5984  df-oprab 5985  df-mpt2 5986  df-fbas 16590  df-fg 16591  df-top 16853  df-topon 16856  df-ntr 16974  df-nei 17052  df-fil 17754  df-flim 17847
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