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Theorem elfg 19444
Description: A condition for elements of a generated filter. (Contributed by Jeff Hankins, 3-Sep-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)
Assertion
Ref Expression
elfg  |-  ( F  e.  ( fBas `  X
)  ->  ( A  e.  ( X filGen F )  <-> 
( A  C_  X  /\  E. x  e.  F  x  C_  A ) ) )
Distinct variable groups:    x, A    x, F
Allowed substitution hint:    X( x)

Proof of Theorem elfg
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 fgval 19443 . . 3  |-  ( F  e.  ( fBas `  X
)  ->  ( X filGen F )  =  {
y  e.  ~P X  |  ( F  i^i  ~P y )  =/=  (/) } )
21eleq2d 2510 . 2  |-  ( F  e.  ( fBas `  X
)  ->  ( A  e.  ( X filGen F )  <-> 
A  e.  { y  e.  ~P X  | 
( F  i^i  ~P y )  =/=  (/) } ) )
3 pweq 3863 . . . . . 6  |-  ( y  =  A  ->  ~P y  =  ~P A
)
43ineq2d 3552 . . . . 5  |-  ( y  =  A  ->  ( F  i^i  ~P y )  =  ( F  i^i  ~P A ) )
54neeq1d 2621 . . . 4  |-  ( y  =  A  ->  (
( F  i^i  ~P y )  =/=  (/)  <->  ( F  i^i  ~P A )  =/=  (/) ) )
65elrab 3117 . . 3  |-  ( A  e.  { y  e. 
~P X  |  ( F  i^i  ~P y
)  =/=  (/) }  <->  ( A  e.  ~P X  /\  ( F  i^i  ~P A )  =/=  (/) ) )
7 elfvdm 5716 . . . . 5  |-  ( F  e.  ( fBas `  X
)  ->  X  e.  dom  fBas )
8 elpw2g 4455 . . . . 5  |-  ( X  e.  dom  fBas  ->  ( A  e.  ~P X  <->  A 
C_  X ) )
97, 8syl 16 . . . 4  |-  ( F  e.  ( fBas `  X
)  ->  ( A  e.  ~P X  <->  A  C_  X
) )
10 elin 3539 . . . . . . . 8  |-  ( x  e.  ( F  i^i  ~P A )  <->  ( x  e.  F  /\  x  e.  ~P A ) )
11 selpw 3867 . . . . . . . . 9  |-  ( x  e.  ~P A  <->  x  C_  A
)
1211anbi2i 694 . . . . . . . 8  |-  ( ( x  e.  F  /\  x  e.  ~P A
)  <->  ( x  e.  F  /\  x  C_  A ) )
1310, 12bitri 249 . . . . . . 7  |-  ( x  e.  ( F  i^i  ~P A )  <->  ( x  e.  F  /\  x  C_  A ) )
1413exbii 1634 . . . . . 6  |-  ( E. x  x  e.  ( F  i^i  ~P A
)  <->  E. x ( x  e.  F  /\  x  C_  A ) )
15 n0 3646 . . . . . 6  |-  ( ( F  i^i  ~P A
)  =/=  (/)  <->  E. x  x  e.  ( F  i^i  ~P A ) )
16 df-rex 2721 . . . . . 6  |-  ( E. x  e.  F  x 
C_  A  <->  E. x
( x  e.  F  /\  x  C_  A ) )
1714, 15, 163bitr4i 277 . . . . 5  |-  ( ( F  i^i  ~P A
)  =/=  (/)  <->  E. x  e.  F  x  C_  A
)
1817a1i 11 . . . 4  |-  ( F  e.  ( fBas `  X
)  ->  ( ( F  i^i  ~P A )  =/=  (/)  <->  E. x  e.  F  x  C_  A ) )
199, 18anbi12d 710 . . 3  |-  ( F  e.  ( fBas `  X
)  ->  ( ( A  e.  ~P X  /\  ( F  i^i  ~P A )  =/=  (/) )  <->  ( A  C_  X  /\  E. x  e.  F  x  C_  A
) ) )
206, 19syl5bb 257 . 2  |-  ( F  e.  ( fBas `  X
)  ->  ( A  e.  { y  e.  ~P X  |  ( F  i^i  ~P y )  =/=  (/) }  <->  ( A  C_  X  /\  E. x  e.  F  x  C_  A
) ) )
212, 20bitrd 253 1  |-  ( F  e.  ( fBas `  X
)  ->  ( A  e.  ( X filGen F )  <-> 
( A  C_  X  /\  E. x  e.  F  x  C_  A ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369   E.wex 1586    e. wcel 1756    =/= wne 2606   E.wrex 2716   {crab 2719    i^i cin 3327    C_ wss 3328   (/)c0 3637   ~Pcpw 3860   dom cdm 4840   ` cfv 5418  (class class class)co 6091   fBascfbas 17804   filGencfg 17805
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-sbc 3187  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-br 4293  df-opab 4351  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-iota 5381  df-fun 5420  df-fv 5426  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-fg 17815
This theorem is referenced by:  ssfg  19445  fgss  19446  fgss2  19447  fgfil  19448  elfilss  19449  fgcl  19451  fgabs  19452  fgtr  19463  trfg  19464  uffix  19494  elfm  19520  elfm2  19521  elfm3  19523  fbflim  19549  flffbas  19568  fclsbas  19594  isucn2  19854  metustOLD  20142  metust  20143  cfilucfilOLD  20144  cfilucfil  20145  metuelOLD  20152  metuel  20153  fgcfil  20782  fgmin  28591  filnetlem4  28602
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