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Theorem elfg 20457
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 20456 . . 3  |-  ( F  e.  ( fBas `  X
)  ->  ( X filGen F )  =  {
y  e.  ~P X  |  ( F  i^i  ~P y )  =/=  (/) } )
21eleq2d 2452 . 2  |-  ( F  e.  ( fBas `  X
)  ->  ( A  e.  ( X filGen F )  <-> 
A  e.  { y  e.  ~P X  | 
( F  i^i  ~P y )  =/=  (/) } ) )
3 pweq 3930 . . . . . 6  |-  ( y  =  A  ->  ~P y  =  ~P A
)
43ineq2d 3614 . . . . 5  |-  ( y  =  A  ->  ( F  i^i  ~P y )  =  ( F  i^i  ~P A ) )
54neeq1d 2659 . . . 4  |-  ( y  =  A  ->  (
( F  i^i  ~P y )  =/=  (/)  <->  ( F  i^i  ~P A )  =/=  (/) ) )
65elrab 3182 . . 3  |-  ( A  e.  { y  e. 
~P X  |  ( F  i^i  ~P y
)  =/=  (/) }  <->  ( A  e.  ~P X  /\  ( F  i^i  ~P A )  =/=  (/) ) )
7 elfvdm 5800 . . . . 5  |-  ( F  e.  ( fBas `  X
)  ->  X  e.  dom  fBas )
8 elpw2g 4528 . . . . 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 3601 . . . . . . . 8  |-  ( x  e.  ( F  i^i  ~P A )  <->  ( x  e.  F  /\  x  e.  ~P A ) )
11 selpw 3934 . . . . . . . . 9  |-  ( x  e.  ~P A  <->  x  C_  A
)
1211anbi2i 692 . . . . . . . 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 1675 . . . . . 6  |-  ( E. x  x  e.  ( F  i^i  ~P A
)  <->  E. x ( x  e.  F  /\  x  C_  A ) )
15 n0 3721 . . . . . 6  |-  ( ( F  i^i  ~P A
)  =/=  (/)  <->  E. x  x  e.  ( F  i^i  ~P A ) )
16 df-rex 2738 . . . . . 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 708 . . 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 367    = wceq 1399   E.wex 1620    e. wcel 1826    =/= wne 2577   E.wrex 2733   {crab 2736    i^i cin 3388    C_ wss 3389   (/)c0 3711   ~Pcpw 3927   dom cdm 4913   ` cfv 5496  (class class class)co 6196   fBascfbas 18519   filGencfg 18520
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-8 1828  ax-9 1830  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360  ax-sep 4488  ax-nul 4496  ax-pow 4543  ax-pr 4601
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-eu 2222  df-mo 2223  df-clab 2368  df-cleq 2374  df-clel 2377  df-nfc 2532  df-ne 2579  df-ral 2737  df-rex 2738  df-rab 2741  df-v 3036  df-sbc 3253  df-dif 3392  df-un 3394  df-in 3396  df-ss 3403  df-nul 3712  df-if 3858  df-pw 3929  df-sn 3945  df-pr 3947  df-op 3951  df-uni 4164  df-br 4368  df-opab 4426  df-id 4709  df-xp 4919  df-rel 4920  df-cnv 4921  df-co 4922  df-dm 4923  df-iota 5460  df-fun 5498  df-fv 5504  df-ov 6199  df-oprab 6200  df-mpt2 6201  df-fg 18530
This theorem is referenced by:  ssfg  20458  fgss  20459  fgss2  20460  fgfil  20461  elfilss  20462  fgcl  20464  fgabs  20465  fgtr  20476  trfg  20477  uffix  20507  elfm  20533  elfm2  20534  elfm3  20536  fbflim  20562  flffbas  20581  fclsbas  20607  isucn2  20867  metustOLD  21155  metust  21156  cfilucfilOLD  21157  cfilucfil  21158  metuelOLD  21165  metuel  21166  fgcfil  21795  fgmin  30354  filnetlem4  30365
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