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Theorem fgss 19573
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 3520 . . . . 5  |-  ( F 
C_  G  ->  ( E. x  e.  F  x  C_  t  ->  E. x  e.  G  x  C_  t
) )
21anim2d 565 . . . 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 1011 . . 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 19571 . . . 4  |-  ( F  e.  ( fBas `  X
)  ->  ( t  e.  ( X filGen F )  <-> 
( t  C_  X  /\  E. x  e.  F  x  C_  t ) ) )
543ad2ant1 1009 . . 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 19571 . . . 4  |-  ( G  e.  ( fBas `  X
)  ->  ( t  e.  ( X filGen G )  <-> 
( t  C_  X  /\  E. x  e.  G  x  C_  t ) ) )
763ad2ant2 1010 . . 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 268 . 2  |-  ( ( F  e.  ( fBas `  X )  /\  G  e.  ( fBas `  X
)  /\  F  C_  G
)  ->  ( t  e.  ( X filGen F )  ->  t  e.  ( X filGen G ) ) )
98ssrdv 3465 1  |-  ( ( F  e.  ( fBas `  X )  /\  G  e.  ( fBas `  X
)  /\  F  C_  G
)  ->  ( X filGen F )  C_  ( X filGen G ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 965    e. wcel 1758   E.wrex 2797    C_ wss 3431   ` cfv 5521  (class class class)co 6195   fBascfbas 17924   filGencfg 17925
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431  ax-sep 4516  ax-nul 4524  ax-pow 4573  ax-pr 4634
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2265  df-mo 2266  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2602  df-ne 2647  df-ral 2801  df-rex 2802  df-rab 2805  df-v 3074  df-sbc 3289  df-dif 3434  df-un 3436  df-in 3438  df-ss 3445  df-nul 3741  df-if 3895  df-pw 3965  df-sn 3981  df-pr 3983  df-op 3987  df-uni 4195  df-br 4396  df-opab 4454  df-id 4739  df-xp 4949  df-rel 4950  df-cnv 4951  df-co 4952  df-dm 4953  df-iota 5484  df-fun 5523  df-fv 5529  df-ov 6198  df-oprab 6199  df-mpt2 6200  df-fg 17935
This theorem is referenced by:  fgabs  19579  fgtr  19590  fmss  19646
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