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Theorem fbasweak 17818
Description: A filter base on any set is also a filter base on any larger set. (Contributed by Stefan O'Rear, 2-Aug-2015.)
Assertion
Ref Expression
fbasweak  |-  ( ( F  e.  ( fBas `  X )  /\  F  C_ 
~P Y  /\  Y  e.  V )  ->  F  e.  ( fBas `  Y
) )

Proof of Theorem fbasweak
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp2 958 . 2  |-  ( ( F  e.  ( fBas `  X )  /\  F  C_ 
~P Y  /\  Y  e.  V )  ->  F  C_ 
~P Y )
2 simp1 957 . . . 4  |-  ( ( F  e.  ( fBas `  X )  /\  F  C_ 
~P Y  /\  Y  e.  V )  ->  F  e.  ( fBas `  X
) )
3 elfvdm 5697 . . . . . 6  |-  ( F  e.  ( fBas `  X
)  ->  X  e.  dom  fBas )
433ad2ant1 978 . . . . 5  |-  ( ( F  e.  ( fBas `  X )  /\  F  C_ 
~P Y  /\  Y  e.  V )  ->  X  e.  dom  fBas )
5 isfbas 17782 . . . . 5  |-  ( X  e.  dom  fBas  ->  ( F  e.  ( fBas `  X )  <->  ( F  C_ 
~P X  /\  ( F  =/=  (/)  /\  (/)  e/  F  /\  A. x  e.  F  A. y  e.  F  ( F  i^i  ~P (
x  i^i  y )
)  =/=  (/) ) ) ) )
64, 5syl 16 . . . 4  |-  ( ( F  e.  ( fBas `  X )  /\  F  C_ 
~P Y  /\  Y  e.  V )  ->  ( F  e.  ( fBas `  X )  <->  ( F  C_ 
~P X  /\  ( F  =/=  (/)  /\  (/)  e/  F  /\  A. x  e.  F  A. y  e.  F  ( F  i^i  ~P (
x  i^i  y )
)  =/=  (/) ) ) ) )
72, 6mpbid 202 . . 3  |-  ( ( F  e.  ( fBas `  X )  /\  F  C_ 
~P Y  /\  Y  e.  V )  ->  ( F  C_  ~P X  /\  ( F  =/=  (/)  /\  (/)  e/  F  /\  A. x  e.  F  A. y  e.  F  ( F  i^i  ~P (
x  i^i  y )
)  =/=  (/) ) ) )
87simprd 450 . 2  |-  ( ( F  e.  ( fBas `  X )  /\  F  C_ 
~P Y  /\  Y  e.  V )  ->  ( F  =/=  (/)  /\  (/)  e/  F  /\  A. x  e.  F  A. y  e.  F  ( F  i^i  ~P (
x  i^i  y )
)  =/=  (/) ) )
9 isfbas 17782 . . 3  |-  ( Y  e.  V  ->  ( F  e.  ( fBas `  Y )  <->  ( F  C_ 
~P Y  /\  ( F  =/=  (/)  /\  (/)  e/  F  /\  A. x  e.  F  A. y  e.  F  ( F  i^i  ~P (
x  i^i  y )
)  =/=  (/) ) ) ) )
1093ad2ant3 980 . 2  |-  ( ( F  e.  ( fBas `  X )  /\  F  C_ 
~P Y  /\  Y  e.  V )  ->  ( F  e.  ( fBas `  Y )  <->  ( F  C_ 
~P Y  /\  ( F  =/=  (/)  /\  (/)  e/  F  /\  A. x  e.  F  A. y  e.  F  ( F  i^i  ~P (
x  i^i  y )
)  =/=  (/) ) ) ) )
111, 8, 10mpbir2and 889 1  |-  ( ( F  e.  ( fBas `  X )  /\  F  C_ 
~P Y  /\  Y  e.  V )  ->  F  e.  ( fBas `  Y
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    /\ w3a 936    e. wcel 1717    =/= wne 2550    e/ wnel 2551   A.wral 2649    i^i cin 3262    C_ wss 3263   (/)c0 3571   ~Pcpw 3742   dom cdm 4818   ` cfv 5394   fBascfbas 16615
This theorem is referenced by:  snfbas  17819  fgabs  17832  fgtr  17843  trfg  17844  ssufl  17871  cfiluweak  18246  cfilresi  19119  cmetss  19138  minveclem4a  19198  minveclem4  19200
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pow 4318  ax-pr 4344
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-nel 2553  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-csb 3195  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-br 4154  df-opab 4208  df-mpt 4209  df-id 4439  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-rn 4829  df-res 4830  df-ima 4831  df-iota 5358  df-fun 5396  df-fv 5402  df-fbas 16623
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