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Theorem fbasweak 17576
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 5570 . . . . . 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 17540 . . . . 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 17540 . . 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 1696    =/= wne 2459    e/ wnel 2460   A.wral 2556    i^i cin 3164    C_ wss 3165   (/)c0 3468   ~Pcpw 3638   dom cdm 4705   ` cfv 5271   fBascfbas 17534
This theorem is referenced by:  snfbas  17577  fgabs  17590  fgtr  17601  trfg  17602  ssufl  17629  cfilresi  18737  cmetss  18756  minveclem4a  18810  minveclem4  18812
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-nel 2462  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  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-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fv 5279  df-fbas 17536
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