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Theorem elfg 20123
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 20122 . . 3  |-  ( F  e.  ( fBas `  X
)  ->  ( X filGen F )  =  {
y  e.  ~P X  |  ( F  i^i  ~P y )  =/=  (/) } )
21eleq2d 2537 . 2  |-  ( F  e.  ( fBas `  X
)  ->  ( A  e.  ( X filGen F )  <-> 
A  e.  { y  e.  ~P X  | 
( F  i^i  ~P y )  =/=  (/) } ) )
3 pweq 4013 . . . . . 6  |-  ( y  =  A  ->  ~P y  =  ~P A
)
43ineq2d 3700 . . . . 5  |-  ( y  =  A  ->  ( F  i^i  ~P y )  =  ( F  i^i  ~P A ) )
54neeq1d 2744 . . . 4  |-  ( y  =  A  ->  (
( F  i^i  ~P y )  =/=  (/)  <->  ( F  i^i  ~P A )  =/=  (/) ) )
65elrab 3261 . . 3  |-  ( A  e.  { y  e. 
~P X  |  ( F  i^i  ~P y
)  =/=  (/) }  <->  ( A  e.  ~P X  /\  ( F  i^i  ~P A )  =/=  (/) ) )
7 elfvdm 5891 . . . . 5  |-  ( F  e.  ( fBas `  X
)  ->  X  e.  dom  fBas )
8 elpw2g 4610 . . . . 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 3687 . . . . . . . 8  |-  ( x  e.  ( F  i^i  ~P A )  <->  ( x  e.  F  /\  x  e.  ~P A ) )
11 selpw 4017 . . . . . . . . 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 1644 . . . . . 6  |-  ( E. x  x  e.  ( F  i^i  ~P A
)  <->  E. x ( x  e.  F  /\  x  C_  A ) )
15 n0 3794 . . . . . 6  |-  ( ( F  i^i  ~P A
)  =/=  (/)  <->  E. x  x  e.  ( F  i^i  ~P A ) )
16 df-rex 2820 . . . . . 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 1379   E.wex 1596    e. wcel 1767    =/= wne 2662   E.wrex 2815   {crab 2818    i^i cin 3475    C_ wss 3476   (/)c0 3785   ~Pcpw 4010   dom cdm 4999   ` cfv 5587  (class class class)co 6283   fBascfbas 18193   filGencfg 18194
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-iota 5550  df-fun 5589  df-fv 5595  df-ov 6286  df-oprab 6287  df-mpt2 6288  df-fg 18204
This theorem is referenced by:  ssfg  20124  fgss  20125  fgss2  20126  fgfil  20127  elfilss  20128  fgcl  20130  fgabs  20131  fgtr  20142  trfg  20143  uffix  20173  elfm  20199  elfm2  20200  elfm3  20202  fbflim  20228  flffbas  20247  fclsbas  20273  isucn2  20533  metustOLD  20821  metust  20822  cfilucfilOLD  20823  cfilucfil  20824  metuelOLD  20831  metuel  20832  fgcfil  21461  fgmin  29807  filnetlem4  29818
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