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Theorem fbflim 20769
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 20671 . . . 4  |-  ( B  e.  ( fBas `  X
)  ->  ( X filGen B )  e.  ( Fil `  X ) )
31, 2syl5eqel 2494 . . 3  |-  ( B  e.  ( fBas `  X
)  ->  F  e.  ( Fil `  X ) )
4 flimopn 20768 . . 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 472 . 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 2480 . . . . . . 7  |-  ( x  e.  F  <->  x  e.  ( X filGen B ) )
7 elfg 20664 . . . . . . . 8  |-  ( B  e.  ( fBas `  X
)  ->  ( x  e.  ( X filGen B )  <-> 
( x  C_  X  /\  E. y  e.  B  y  C_  x ) ) )
87ad3antlr 729 . . . . . . 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 752 . . . . . . . 8  |-  ( ( ( J  e.  (TopOn `  X )  /\  B  e.  ( fBas `  X
) )  /\  A  e.  X )  ->  J  e.  (TopOn `  X )
)
11 toponss 19722 . . . . . . . 8  |-  ( ( J  e.  (TopOn `  X )  /\  x  e.  J )  ->  x  C_  X )
1210, 11sylan 469 . . . . . . 7  |-  ( ( ( ( J  e.  (TopOn `  X )  /\  B  e.  ( fBas `  X ) )  /\  A  e.  X
)  /\  x  e.  J )  ->  x  C_  X )
1312biantrurd 506 . . . . . 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 314 . . . 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 2840 . . 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 639 . 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 367    = wceq 1405    e. wcel 1842   A.wral 2754   E.wrex 2755    C_ wss 3414   ` cfv 5569  (class class class)co 6278   fBascfbas 18726   filGencfg 18727  TopOnctopon 19687   Filcfil 20638    fLim cflim 20727
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-rep 4507  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-nel 2601  df-ral 2759  df-rex 2760  df-reu 2761  df-rab 2763  df-v 3061  df-sbc 3278  df-csb 3374  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4192  df-iun 4273  df-br 4396  df-opab 4454  df-mpt 4455  df-id 4738  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-ov 6281  df-oprab 6282  df-mpt2 6283  df-fbas 18736  df-fg 18737  df-top 19691  df-topon 19694  df-ntr 19813  df-nei 19892  df-fil 20639  df-flim 20732
This theorem is referenced by:  fbflim2  20770
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