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Theorem fgval 19443
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 17815 . . 3  |-  filGen  =  ( v  e.  _V , 
f  e.  ( fBas `  v )  |->  { x  e.  ~P v  |  ( f  i^i  ~P x
)  =/=  (/) } )
21a1i 11 . 2  |-  ( F  e.  ( fBas `  X
)  ->  filGen  =  ( v  e.  _V , 
f  e.  ( fBas `  v )  |->  { x  e.  ~P v  |  ( f  i^i  ~P x
)  =/=  (/) } ) )
3 pweq 3863 . . . . 5  |-  ( v  =  X  ->  ~P v  =  ~P X
)
43adantr 465 . . . 4  |-  ( ( v  =  X  /\  f  =  F )  ->  ~P v  =  ~P X )
5 ineq1 3545 . . . . . 6  |-  ( f  =  F  ->  (
f  i^i  ~P x
)  =  ( F  i^i  ~P x ) )
65neeq1d 2621 . . . . 5  |-  ( f  =  F  ->  (
( f  i^i  ~P x )  =/=  (/)  <->  ( F  i^i  ~P x )  =/=  (/) ) )
76adantl 466 . . . 4  |-  ( ( v  =  X  /\  f  =  F )  ->  ( ( f  i^i 
~P x )  =/=  (/) 
<->  ( F  i^i  ~P x )  =/=  (/) ) )
84, 7rabeqbidv 2967 . . 3  |-  ( ( v  =  X  /\  f  =  F )  ->  { x  e.  ~P v  |  ( f  i^i  ~P x )  =/=  (/) }  =  { x  e.  ~P X  |  ( F  i^i  ~P x
)  =/=  (/) } )
98adantl 466 . 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 5691 . . 3  |-  ( v  =  X  ->  ( fBas `  v )  =  ( fBas `  X
) )
1110adantl 466 . 2  |-  ( ( F  e.  ( fBas `  X )  /\  v  =  X )  ->  ( fBas `  v )  =  ( fBas `  X
) )
12 elfvex 5717 . 2  |-  ( F  e.  ( fBas `  X
)  ->  X  e.  _V )
13 id 22 . 2  |-  ( F  e.  ( fBas `  X
)  ->  F  e.  ( fBas `  X )
)
14 elfvdm 5716 . . 3  |-  ( F  e.  ( fBas `  X
)  ->  X  e.  dom  fBas )
15 pwexg 4476 . . 3  |-  ( X  e.  dom  fBas  ->  ~P X  e.  _V )
16 rabexg 4442 . . 3  |-  ( ~P X  e.  _V  ->  { x  e.  ~P X  |  ( F  i^i  ~P x )  =/=  (/) }  e.  _V )
1714, 15, 163syl 20 . 2  |-  ( F  e.  ( fBas `  X
)  ->  { x  e.  ~P X  |  ( F  i^i  ~P x
)  =/=  (/) }  e.  _V )
182, 9, 11, 12, 13, 17ovmpt2dx 6217 1  |-  ( F  e.  ( fBas `  X
)  ->  ( X filGen F )  =  {
x  e.  ~P X  |  ( F  i^i  ~P x )  =/=  (/) } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1369    e. wcel 1756    =/= wne 2606   {crab 2719   _Vcvv 2972    i^i cin 3327   (/)c0 3637   ~Pcpw 3860   dom cdm 4840   ` cfv 5418  (class class class)co 6091    e. cmpt2 6093   fBascfbas 17804   filGencfg 17805
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pow 4470  ax-pr 4531
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-sbc 3187  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-pw 3862  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-br 4293  df-opab 4351  df-id 4636  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-iota 5381  df-fun 5420  df-fv 5426  df-ov 6094  df-oprab 6095  df-mpt2 6096  df-fg 17815
This theorem is referenced by:  elfg  19444  restmetu  20162  neifg  28592
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