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Theorem fgss 17907
Description: A bigger base generates a bigger filter. (Contributed by NM, 5-Sep-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)
Assertion
Ref Expression
fgss  |-  ( ( F  e.  ( fBas `  X )  /\  G  e.  ( fBas `  X
)  /\  F  C_  G
)  ->  ( X filGen F )  C_  ( X filGen G ) )

Proof of Theorem fgss
Dummy variables  x  t are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssrexv 3410 . . . . 5  |-  ( F 
C_  G  ->  ( E. x  e.  F  x  C_  t  ->  E. x  e.  G  x  C_  t
) )
21anim2d 550 . . . 4  |-  ( F 
C_  G  ->  (
( t  C_  X  /\  E. x  e.  F  x  C_  t )  -> 
( t  C_  X  /\  E. x  e.  G  x  C_  t ) ) )
323ad2ant3 981 . . 3  |-  ( ( F  e.  ( fBas `  X )  /\  G  e.  ( fBas `  X
)  /\  F  C_  G
)  ->  ( (
t  C_  X  /\  E. x  e.  F  x 
C_  t )  -> 
( t  C_  X  /\  E. x  e.  G  x  C_  t ) ) )
4 elfg 17905 . . . 4  |-  ( F  e.  ( fBas `  X
)  ->  ( t  e.  ( X filGen F )  <-> 
( t  C_  X  /\  E. x  e.  F  x  C_  t ) ) )
543ad2ant1 979 . . 3  |-  ( ( F  e.  ( fBas `  X )  /\  G  e.  ( fBas `  X
)  /\  F  C_  G
)  ->  ( t  e.  ( X filGen F )  <-> 
( t  C_  X  /\  E. x  e.  F  x  C_  t ) ) )
6 elfg 17905 . . . 4  |-  ( G  e.  ( fBas `  X
)  ->  ( t  e.  ( X filGen G )  <-> 
( t  C_  X  /\  E. x  e.  G  x  C_  t ) ) )
763ad2ant2 980 . . 3  |-  ( ( F  e.  ( fBas `  X )  /\  G  e.  ( fBas `  X
)  /\  F  C_  G
)  ->  ( t  e.  ( X filGen G )  <-> 
( t  C_  X  /\  E. x  e.  G  x  C_  t ) ) )
83, 5, 73imtr4d 261 . 2  |-  ( ( F  e.  ( fBas `  X )  /\  G  e.  ( fBas `  X
)  /\  F  C_  G
)  ->  ( t  e.  ( X filGen F )  ->  t  e.  ( X filGen G ) ) )
98ssrdv 3356 1  |-  ( ( F  e.  ( fBas `  X )  /\  G  e.  ( fBas `  X
)  /\  F  C_  G
)  ->  ( X filGen F )  C_  ( X filGen G ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    /\ w3a 937    e. wcel 1726   E.wrex 2708    C_ wss 3322   ` cfv 5456  (class class class)co 6083   fBascfbas 16691   filGencfg 16692
This theorem is referenced by:  fgabs  17913  fgtr  17924  fmss  17980
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pow 4379  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4215  df-opab 4269  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-iota 5420  df-fun 5458  df-fv 5464  df-ov 6086  df-oprab 6087  df-mpt2 6088  df-fg 16702
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