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Theorem fbflim 17969
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 17871 . . . 4  |-  ( B  e.  ( fBas `  X
)  ->  ( X filGen B )  e.  ( Fil `  X ) )
31, 2syl5eqel 2496 . . 3  |-  ( B  e.  ( fBas `  X
)  ->  F  e.  ( Fil `  X ) )
4 flimopn 17968 . . 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 461 . 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 2476 . . . . . . 7  |-  ( x  e.  F  <->  x  e.  ( X filGen B ) )
7 elfg 17864 . . . . . . . 8  |-  ( B  e.  ( fBas `  X
)  ->  ( x  e.  ( X filGen B )  <-> 
( x  C_  X  /\  E. y  e.  B  y  C_  x ) ) )
87ad3antlr 712 . . . . . . 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 ) ) )
96, 8syl5bb 249 . . . . . 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 ) ) )
10 simpll 731 . . . . . . . 8  |-  ( ( ( J  e.  (TopOn `  X )  /\  B  e.  ( fBas `  X
) )  /\  A  e.  X )  ->  J  e.  (TopOn `  X )
)
11 toponss 16957 . . . . . . . 8  |-  ( ( J  e.  (TopOn `  X )  /\  x  e.  J )  ->  x  C_  X )
1210, 11sylan 458 . . . . . . 7  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  B  e.  ( fBas `  X ) )  /\  A  e.  X
)  /\  x  e.  J )  ->  x  C_  X )
1312biantrurd 495 . . . . . 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 ) ) )
149, 13bitr4d 248 . . . . 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 ) )
1514imbi2d 308 . . . 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 ) ) )
1615ralbidva 2690 . . 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 ) ) )
1716pm5.32da 623 . 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 ) ) ) )
185, 17bitrd 245 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 177    /\ wa 359    = wceq 1649    e. wcel 1721   A.wral 2674   E.wrex 2675    C_ wss 3288   ` cfv 5421  (class class class)co 6048   fBascfbas 16652   filGencfg 16653  TopOnctopon 16922   Filcfil 17838    fLim cflim 17927
This theorem is referenced by:  fbflim2  17970
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-rep 4288  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-nel 2578  df-ral 2679  df-rex 2680  df-reu 2681  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-f1 5426  df-fo 5427  df-f1o 5428  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-fbas 16662  df-fg 16663  df-top 16926  df-topon 16929  df-ntr 17047  df-nei 17125  df-fil 17839  df-flim 17932
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