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Theorem fgval 17581
Description: The filter generating class gives a filter for every filter base. (Contributed by Jeff Hankins, 3-Sep-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)
Assertion
Ref Expression
fgval  |-  ( F  e.  ( fBas `  X
)  ->  ( X filGen F )  =  {
x  e.  ~P X  |  ( F  i^i  ~P x )  =/=  (/) } )
Distinct variable groups:    x, F    x, X

Proof of Theorem fgval
Dummy variables  v 
f are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-fg 17537 . . 3  |-  filGen  =  ( v  e.  _V , 
f  e.  ( fBas `  v )  |->  { x  e.  ~P v  |  ( f  i^i  ~P x
)  =/=  (/) } )
21a1i 10 . 2  |-  ( F  e.  ( fBas `  X
)  ->  filGen  =  ( v  e.  _V , 
f  e.  ( fBas `  v )  |->  { x  e.  ~P v  |  ( f  i^i  ~P x
)  =/=  (/) } ) )
3 pweq 3641 . . . . 5  |-  ( v  =  X  ->  ~P v  =  ~P X
)
43adantr 451 . . . 4  |-  ( ( v  =  X  /\  f  =  F )  ->  ~P v  =  ~P X )
5 ineq1 3376 . . . . . 6  |-  ( f  =  F  ->  (
f  i^i  ~P x
)  =  ( F  i^i  ~P x ) )
65neeq1d 2472 . . . . 5  |-  ( f  =  F  ->  (
( f  i^i  ~P x )  =/=  (/)  <->  ( F  i^i  ~P x )  =/=  (/) ) )
76adantl 452 . . . 4  |-  ( ( v  =  X  /\  f  =  F )  ->  ( ( f  i^i 
~P x )  =/=  (/) 
<->  ( F  i^i  ~P x )  =/=  (/) ) )
84, 7rabeqbidv 2796 . . 3  |-  ( ( v  =  X  /\  f  =  F )  ->  { x  e.  ~P v  |  ( f  i^i  ~P x )  =/=  (/) }  =  { x  e.  ~P X  |  ( F  i^i  ~P x
)  =/=  (/) } )
98adantl 452 . 2  |-  ( ( F  e.  ( fBas `  X )  /\  (
v  =  X  /\  f  =  F )
)  ->  { x  e.  ~P v  |  ( f  i^i  ~P x
)  =/=  (/) }  =  { x  e.  ~P X  |  ( F  i^i  ~P x )  =/=  (/) } )
10 fveq2 5541 . . 3  |-  ( v  =  X  ->  ( fBas `  v )  =  ( fBas `  X
) )
1110adantl 452 . 2  |-  ( ( F  e.  ( fBas `  X )  /\  v  =  X )  ->  ( fBas `  v )  =  ( fBas `  X
) )
12 elfvex 5571 . 2  |-  ( F  e.  ( fBas `  X
)  ->  X  e.  _V )
13 id 19 . 2  |-  ( F  e.  ( fBas `  X
)  ->  F  e.  ( fBas `  X )
)
14 elfvdm 5570 . . 3  |-  ( F  e.  ( fBas `  X
)  ->  X  e.  dom  fBas )
15 pwexg 4210 . . 3  |-  ( X  e.  dom  fBas  ->  ~P X  e.  _V )
16 rabexg 4180 . . 3  |-  ( ~P X  e.  _V  ->  { x  e.  ~P X  |  ( F  i^i  ~P x )  =/=  (/) }  e.  _V )
1714, 15, 163syl 18 . 2  |-  ( F  e.  ( fBas `  X
)  ->  { x  e.  ~P X  |  ( F  i^i  ~P x
)  =/=  (/) }  e.  _V )
182, 9, 11, 12, 13, 17ovmpt2dx 5990 1  |-  ( F  e.  ( fBas `  X
)  ->  ( X filGen F )  =  {
x  e.  ~P X  |  ( F  i^i  ~P x )  =/=  (/) } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696    =/= wne 2459   {crab 2560   _Vcvv 2801    i^i cin 3164   (/)c0 3468   ~Pcpw 3638   dom cdm 4705   ` cfv 5271  (class class class)co 5874    e. cmpt2 5876   fBascfbas 17534   filGencfg 17535
This theorem is referenced by:  elfg  17582  neifg  26423
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-fg 17537
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