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Theorem elfg 17856
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 17855 . . 3  |-  ( F  e.  ( fBas `  X
)  ->  ( X filGen F )  =  {
y  e.  ~P X  |  ( F  i^i  ~P y )  =/=  (/) } )
21eleq2d 2471 . 2  |-  ( F  e.  ( fBas `  X
)  ->  ( A  e.  ( X filGen F )  <-> 
A  e.  { y  e.  ~P X  | 
( F  i^i  ~P y )  =/=  (/) } ) )
3 pweq 3762 . . . . . 6  |-  ( y  =  A  ->  ~P y  =  ~P A
)
43ineq2d 3502 . . . . 5  |-  ( y  =  A  ->  ( F  i^i  ~P y )  =  ( F  i^i  ~P A ) )
54neeq1d 2580 . . . 4  |-  ( y  =  A  ->  (
( F  i^i  ~P y )  =/=  (/)  <->  ( F  i^i  ~P A )  =/=  (/) ) )
65elrab 3052 . . 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 4323 . . . . 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 3490 . . . . . . . 8  |-  ( x  e.  ( F  i^i  ~P A )  <->  ( x  e.  F  /\  x  e.  ~P A ) )
11 vex 2919 . . . . . . . . . 10  |-  x  e. 
_V
1211elpw 3765 . . . . . . . . 9  |-  ( x  e.  ~P A  <->  x  C_  A
)
1312anbi2i 676 . . . . . . . 8  |-  ( ( x  e.  F  /\  x  e.  ~P A
)  <->  ( x  e.  F  /\  x  C_  A ) )
1410, 13bitri 241 . . . . . . 7  |-  ( x  e.  ( F  i^i  ~P A )  <->  ( x  e.  F  /\  x  C_  A ) )
1514exbii 1589 . . . . . 6  |-  ( E. x  x  e.  ( F  i^i  ~P A
)  <->  E. x ( x  e.  F  /\  x  C_  A ) )
16 n0 3597 . . . . . 6  |-  ( ( F  i^i  ~P A
)  =/=  (/)  <->  E. x  x  e.  ( F  i^i  ~P A ) )
17 df-rex 2672 . . . . . 6  |-  ( E. x  e.  F  x 
C_  A  <->  E. x
( x  e.  F  /\  x  C_  A ) )
1815, 16, 173bitr4i 269 . . . . 5  |-  ( ( F  i^i  ~P A
)  =/=  (/)  <->  E. x  e.  F  x  C_  A
)
1918a1i 11 . . . 4  |-  ( F  e.  ( fBas `  X
)  ->  ( ( F  i^i  ~P A )  =/=  (/)  <->  E. x  e.  F  x  C_  A ) )
209, 19anbi12d 692 . . 3  |-  ( F  e.  ( fBas `  X
)  ->  ( ( A  e.  ~P X  /\  ( F  i^i  ~P A )  =/=  (/) )  <->  ( A  C_  X  /\  E. x  e.  F  x  C_  A
) ) )
216, 20syl5bb 249 . 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
) ) )
222, 21bitrd 245 1  |-  ( F  e.  ( fBas `  X
)  ->  ( A  e.  ( X filGen F )  <-> 
( A  C_  X  /\  E. x  e.  F  x  C_  A ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359   E.wex 1547    = wceq 1649    e. wcel 1721    =/= wne 2567   E.wrex 2667   {crab 2670    i^i cin 3279    C_ wss 3280   (/)c0 3588   ~Pcpw 3759   dom cdm 4837   ` cfv 5413  (class class class)co 6040   fBascfbas 16644   filGencfg 16645
This theorem is referenced by:  ssfg  17857  fgss  17858  fgss2  17859  fgfil  17860  elfilss  17861  fgcl  17863  fgabs  17864  fgtr  17875  trfg  17876  uffix  17906  elfm  17932  elfm2  17933  elfm3  17935  fbflim  17961  flffbas  17980  fclsbas  18006  isucn2  18262  metustOLD  18550  metust  18551  cfilucfilOLD  18552  cfilucfil  18553  metuelOLD  18560  metuel  18561  fgcfil  19177  fgmin  26289  filnetlem4  26300
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 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-iota 5377  df-fun 5415  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-fg 16655
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