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Theorem fbflim 17671
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 17573 . . . 4  |-  ( B  e.  ( fBas `  X
)  ->  ( X filGen B )  e.  ( Fil `  X ) )
31, 2syl5eqel 2367 . . 3  |-  ( B  e.  ( fBas `  X
)  ->  F  e.  ( Fil `  X ) )
4 flimopn 17670 . . 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 2347 . . . . . . 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 17566 . . . . . . . 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 16667 . . . . . . . 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 2559 . . 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 1623    e. wcel 1684   A.wral 2543   E.wrex 2544    C_ wss 3152   ` cfv 5255  (class class class)co 5858  TopOnctopon 16632   fBascfbas 17518   filGencfg 17519   Filcfil 17540    fLim cflim 17629
This theorem is referenced by:  fbflim2  17672
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-rep 4131  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-top 16636  df-topon 16639  df-ntr 16757  df-nei 16835  df-fbas 17520  df-fg 17521  df-fil 17541  df-flim 17634
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