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Theorem fgss 17568
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 3238 . . . . 5  |-  ( F 
C_  G  ->  ( E. x  e.  F  x  C_  t  ->  E. x  e.  G  x  C_  t
) )
21anim2d 548 . . . 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 978 . . 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 17566 . . . 4  |-  ( F  e.  ( fBas `  X
)  ->  ( t  e.  ( X filGen F )  <-> 
( t  C_  X  /\  E. x  e.  F  x  C_  t ) ) )
543ad2ant1 976 . . 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 17566 . . . 4  |-  ( G  e.  ( fBas `  X
)  ->  ( t  e.  ( X filGen G )  <-> 
( t  C_  X  /\  E. x  e.  G  x  C_  t ) ) )
763ad2ant2 977 . . 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 259 . 2  |-  ( ( F  e.  ( fBas `  X )  /\  G  e.  ( fBas `  X
)  /\  F  C_  G
)  ->  ( t  e.  ( X filGen F )  ->  t  e.  ( X filGen G ) ) )
98ssrdv 3185 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 176    /\ wa 358    /\ w3a 934    e. wcel 1684   E.wrex 2544    C_ wss 3152   ` cfv 5255  (class class class)co 5858   fBascfbas 17518   filGencfg 17519
This theorem is referenced by:  fgabs  17574  fgtr  17585  fmss  17641
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-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  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-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-fg 17521
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