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Theorem elfg 20964
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 20963 . . 3  |-  ( F  e.  ( fBas `  X
)  ->  ( X filGen F )  =  {
y  e.  ~P X  |  ( F  i^i  ~P y )  =/=  (/) } )
21eleq2d 2534 . 2  |-  ( F  e.  ( fBas `  X
)  ->  ( A  e.  ( X filGen F )  <-> 
A  e.  { y  e.  ~P X  | 
( F  i^i  ~P y )  =/=  (/) } ) )
3 pweq 3945 . . . . . 6  |-  ( y  =  A  ->  ~P y  =  ~P A
)
43ineq2d 3625 . . . . 5  |-  ( y  =  A  ->  ( F  i^i  ~P y )  =  ( F  i^i  ~P A ) )
54neeq1d 2702 . . . 4  |-  ( y  =  A  ->  (
( F  i^i  ~P y )  =/=  (/)  <->  ( F  i^i  ~P A )  =/=  (/) ) )
65elrab 3184 . . 3  |-  ( A  e.  { y  e. 
~P X  |  ( F  i^i  ~P y
)  =/=  (/) }  <->  ( A  e.  ~P X  /\  ( F  i^i  ~P A )  =/=  (/) ) )
7 elfvdm 5905 . . . . 5  |-  ( F  e.  ( fBas `  X
)  ->  X  e.  dom  fBas )
8 elpw2g 4564 . . . . 5  |-  ( X  e.  dom  fBas  ->  ( A  e.  ~P X  <->  A 
C_  X ) )
97, 8syl 17 . . . 4  |-  ( F  e.  ( fBas `  X
)  ->  ( A  e.  ~P X  <->  A  C_  X
) )
10 elin 3608 . . . . . . . 8  |-  ( x  e.  ( F  i^i  ~P A )  <->  ( x  e.  F  /\  x  e.  ~P A ) )
11 selpw 3949 . . . . . . . . 9  |-  ( x  e.  ~P A  <->  x  C_  A
)
1211anbi2i 708 . . . . . . . 8  |-  ( ( x  e.  F  /\  x  e.  ~P A
)  <->  ( x  e.  F  /\  x  C_  A ) )
1310, 12bitri 257 . . . . . . 7  |-  ( x  e.  ( F  i^i  ~P A )  <->  ( x  e.  F  /\  x  C_  A ) )
1413exbii 1726 . . . . . 6  |-  ( E. x  x  e.  ( F  i^i  ~P A
)  <->  E. x ( x  e.  F  /\  x  C_  A ) )
15 n0 3732 . . . . . 6  |-  ( ( F  i^i  ~P A
)  =/=  (/)  <->  E. x  x  e.  ( F  i^i  ~P A ) )
16 df-rex 2762 . . . . . 6  |-  ( E. x  e.  F  x 
C_  A  <->  E. x
( x  e.  F  /\  x  C_  A ) )
1714, 15, 163bitr4i 285 . . . . 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 725 . . 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 265 . 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 261 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 189    /\ wa 376    = wceq 1452   E.wex 1671    e. wcel 1904    =/= wne 2641   E.wrex 2757   {crab 2760    i^i cin 3389    C_ wss 3390   (/)c0 3722   ~Pcpw 3942   dom cdm 4839   ` cfv 5589  (class class class)co 6308   fBascfbas 19035   filGencfg 19036
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-iota 5553  df-fun 5591  df-fv 5597  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-fg 19045
This theorem is referenced by:  ssfg  20965  fgss  20966  fgss2  20967  fgfil  20968  elfilss  20969  fgcl  20971  fgabs  20972  fgtr  20983  trfg  20984  uffix  21014  elfm  21040  elfm2  21041  elfm3  21043  fbflim  21069  flffbas  21088  fclsbas  21114  isucn2  21372  metust  21651  cfilucfil  21652  metuel  21657  fgcfil  22319  fgmin  31097  filnetlem4  31108
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