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Theorem fbasweak 17560
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 956 . 2  |-  ( ( F  e.  ( fBas `  X )  /\  F  C_ 
~P Y  /\  Y  e.  V )  ->  F  C_ 
~P Y )
2 simp1 955 . . . 4  |-  ( ( F  e.  ( fBas `  X )  /\  F  C_ 
~P Y  /\  Y  e.  V )  ->  F  e.  ( fBas `  X
) )
3 elfvdm 5554 . . . . . 6  |-  ( F  e.  ( fBas `  X
)  ->  X  e.  dom  fBas )
433ad2ant1 976 . . . . 5  |-  ( ( F  e.  ( fBas `  X )  /\  F  C_ 
~P Y  /\  Y  e.  V )  ->  X  e.  dom  fBas )
5 isfbas 17524 . . . . 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 15 . . . 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 201 . . 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 449 . 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 17524 . . 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 978 . 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 888 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 176    /\ wa 358    /\ w3a 934    e. wcel 1684    =/= wne 2446    e/ wnel 2447   A.wral 2543    i^i cin 3151    C_ wss 3152   (/)c0 3455   ~Pcpw 3625   dom cdm 4689   ` cfv 5255   fBascfbas 17518
This theorem is referenced by:  snfbas  17561  fgabs  17574  fgtr  17585  trfg  17586  ssufl  17613  cfilresi  18721  cmetss  18740  minveclem4a  18794  minveclem4  18796
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-nel 2449  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fv 5263  df-fbas 17520
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