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Theorem fgss 17858
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 3368 . . . . 5  |-  ( F 
C_  G  ->  ( E. x  e.  F  x  C_  t  ->  E. x  e.  G  x  C_  t
) )
21anim2d 549 . . . 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 980 . . 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 17856 . . . 4  |-  ( F  e.  ( fBas `  X
)  ->  ( t  e.  ( X filGen F )  <-> 
( t  C_  X  /\  E. x  e.  F  x  C_  t ) ) )
543ad2ant1 978 . . 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 17856 . . . 4  |-  ( G  e.  ( fBas `  X
)  ->  ( t  e.  ( X filGen G )  <-> 
( t  C_  X  /\  E. x  e.  G  x  C_  t ) ) )
763ad2ant2 979 . . 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 260 . 2  |-  ( ( F  e.  ( fBas `  X )  /\  G  e.  ( fBas `  X
)  /\  F  C_  G
)  ->  ( t  e.  ( X filGen F )  ->  t  e.  ( X filGen G ) ) )
98ssrdv 3314 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 177    /\ wa 359    /\ w3a 936    e. wcel 1721   E.wrex 2667    C_ wss 3280   ` cfv 5413  (class class class)co 6040   fBascfbas 16644   filGencfg 16645
This theorem is referenced by:  fgabs  17864  fgtr  17875  fmss  17931
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pow 4337  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-iota 5377  df-fun 5415  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-fg 16655
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