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Theorem fgss 17664
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 3314 . . . . 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 17662 . . . 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 17662 . . . 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 3261 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 1710   E.wrex 2620    C_ wss 3228   ` cfv 5334  (class class class)co 5942   fBascfbas 16465   filGencfg 16466
This theorem is referenced by:  fgabs  17670  fgtr  17681  fmss  17737
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1930  ax-ext 2339  ax-sep 4220  ax-nul 4228  ax-pow 4267  ax-pr 4293
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2213  df-mo 2214  df-clab 2345  df-cleq 2351  df-clel 2354  df-nfc 2483  df-ne 2523  df-ral 2624  df-rex 2625  df-rab 2628  df-v 2866  df-sbc 3068  df-dif 3231  df-un 3233  df-in 3235  df-ss 3242  df-nul 3532  df-if 3642  df-pw 3703  df-sn 3722  df-pr 3723  df-op 3725  df-uni 3907  df-br 4103  df-opab 4157  df-id 4388  df-xp 4774  df-rel 4775  df-cnv 4776  df-co 4777  df-dm 4778  df-iota 5298  df-fun 5336  df-fv 5342  df-ov 5945  df-oprab 5946  df-mpt2 5947  df-fg 16474
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