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Theorem fbflim 20205
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 20107 . . . 4  |-  ( B  e.  ( fBas `  X
)  ->  ( X filGen B )  e.  ( Fil `  X ) )
31, 2syl5eqel 2552 . . 3  |-  ( B  e.  ( fBas `  X
)  ->  F  e.  ( Fil `  X ) )
4 flimopn 20204 . . 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 474 . 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 2538 . . . . . . 7  |-  ( x  e.  F  <->  x  e.  ( X filGen B ) )
7 elfg 20100 . . . . . . . 8  |-  ( B  e.  ( fBas `  X
)  ->  ( x  e.  ( X filGen B )  <-> 
( x  C_  X  /\  E. y  e.  B  y  C_  x ) ) )
87ad3antlr 730 . . . . . . 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 257 . . . . . 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 753 . . . . . . . 8  |-  ( ( ( J  e.  (TopOn `  X )  /\  B  e.  ( fBas `  X
) )  /\  A  e.  X )  ->  J  e.  (TopOn `  X )
)
11 toponss 19190 . . . . . . . 8  |-  ( ( J  e.  (TopOn `  X )  /\  x  e.  J )  ->  x  C_  X )
1210, 11sylan 471 . . . . . . 7  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  B  e.  ( fBas `  X ) )  /\  A  e.  X
)  /\  x  e.  J )  ->  x  C_  X )
1312biantrurd 508 . . . . . 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 256 . . . . 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 316 . . . 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 2893 . . 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 641 . 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 253 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 setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1374    e. wcel 1762   A.wral 2807   E.wrex 2808    C_ wss 3469   ` cfv 5579  (class class class)co 6275   fBascfbas 18170   filGencfg 18171  TopOnctopon 19155   Filcfil 20074    fLim cflim 20163
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-fbas 18180  df-fg 18181  df-top 19159  df-topon 19162  df-ntr 19280  df-nei 19358  df-fil 20075  df-flim 20168
This theorem is referenced by:  fbflim2  20206
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