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Theorem fgmin 26400
Description: Minimality property of a generated filter: every filter that contains  B contains its generated filter. (Contributed by Jeff Hankins, 5-Sep-2009.) (Revised by Mario Carneiro, 7-Aug-2015.)
Assertion
Ref Expression
fgmin  |-  ( ( B  e.  ( fBas `  X )  /\  F  e.  ( Fil `  X
) )  ->  ( B  C_  F  <->  ( X filGen B )  C_  F
) )

Proof of Theorem fgmin
Dummy variables  x  t are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elfg 17904 . . . . . . 7  |-  ( B  e.  ( fBas `  X
)  ->  ( t  e.  ( X filGen B )  <-> 
( t  C_  X  /\  E. x  e.  B  x  C_  t ) ) )
21adantr 453 . . . . . 6  |-  ( ( B  e.  ( fBas `  X )  /\  F  e.  ( Fil `  X
) )  ->  (
t  e.  ( X
filGen B )  <->  ( t  C_  X  /\  E. x  e.  B  x  C_  t
) ) )
32adantr 453 . . . . 5  |-  ( ( ( B  e.  (
fBas `  X )  /\  F  e.  ( Fil `  X ) )  /\  B  C_  F
)  ->  ( t  e.  ( X filGen B )  <-> 
( t  C_  X  /\  E. x  e.  B  x  C_  t ) ) )
4 ssrexv 3409 . . . . . . . . 9  |-  ( B 
C_  F  ->  ( E. x  e.  B  x  C_  t  ->  E. x  e.  F  x  C_  t
) )
54adantl 454 . . . . . . . 8  |-  ( ( ( B  e.  (
fBas `  X )  /\  F  e.  ( Fil `  X ) )  /\  B  C_  F
)  ->  ( E. x  e.  B  x  C_  t  ->  E. x  e.  F  x  C_  t
) )
6 filss 17886 . . . . . . . . . . . 12  |-  ( ( F  e.  ( Fil `  X )  /\  (
x  e.  F  /\  t  C_  X  /\  x  C_  t ) )  -> 
t  e.  F )
763exp2 1172 . . . . . . . . . . 11  |-  ( F  e.  ( Fil `  X
)  ->  ( x  e.  F  ->  ( t 
C_  X  ->  (
x  C_  t  ->  t  e.  F ) ) ) )
87com34 80 . . . . . . . . . 10  |-  ( F  e.  ( Fil `  X
)  ->  ( x  e.  F  ->  ( x 
C_  t  ->  (
t  C_  X  ->  t  e.  F ) ) ) )
98rexlimdv 2830 . . . . . . . . 9  |-  ( F  e.  ( Fil `  X
)  ->  ( E. x  e.  F  x  C_  t  ->  ( t  C_  X  ->  t  e.  F ) ) )
109ad2antlr 709 . . . . . . . 8  |-  ( ( ( B  e.  (
fBas `  X )  /\  F  e.  ( Fil `  X ) )  /\  B  C_  F
)  ->  ( E. x  e.  F  x  C_  t  ->  ( t  C_  X  ->  t  e.  F ) ) )
115, 10syld 43 . . . . . . 7  |-  ( ( ( B  e.  (
fBas `  X )  /\  F  e.  ( Fil `  X ) )  /\  B  C_  F
)  ->  ( E. x  e.  B  x  C_  t  ->  ( t  C_  X  ->  t  e.  F ) ) )
1211com23 75 . . . . . 6  |-  ( ( ( B  e.  (
fBas `  X )  /\  F  e.  ( Fil `  X ) )  /\  B  C_  F
)  ->  ( t  C_  X  ->  ( E. x  e.  B  x  C_  t  ->  t  e.  F ) ) )
1312imp3a 422 . . . . 5  |-  ( ( ( B  e.  (
fBas `  X )  /\  F  e.  ( Fil `  X ) )  /\  B  C_  F
)  ->  ( (
t  C_  X  /\  E. x  e.  B  x 
C_  t )  -> 
t  e.  F ) )
143, 13sylbid 208 . . . 4  |-  ( ( ( B  e.  (
fBas `  X )  /\  F  e.  ( Fil `  X ) )  /\  B  C_  F
)  ->  ( t  e.  ( X filGen B )  ->  t  e.  F
) )
1514ssrdv 3355 . . 3  |-  ( ( ( B  e.  (
fBas `  X )  /\  F  e.  ( Fil `  X ) )  /\  B  C_  F
)  ->  ( X filGen B )  C_  F
)
1615ex 425 . 2  |-  ( ( B  e.  ( fBas `  X )  /\  F  e.  ( Fil `  X
) )  ->  ( B  C_  F  ->  ( X filGen B )  C_  F ) )
17 ssfg 17905 . . . 4  |-  ( B  e.  ( fBas `  X
)  ->  B  C_  ( X filGen B ) )
18 sstr2 3356 . . . 4  |-  ( B 
C_  ( X filGen B )  ->  ( ( X filGen B )  C_  F  ->  B  C_  F
) )
1917, 18syl 16 . . 3  |-  ( B  e.  ( fBas `  X
)  ->  ( ( X filGen B )  C_  F  ->  B  C_  F
) )
2019adantr 453 . 2  |-  ( ( B  e.  ( fBas `  X )  /\  F  e.  ( Fil `  X
) )  ->  (
( X filGen B ) 
C_  F  ->  B  C_  F ) )
2116, 20impbid 185 1  |-  ( ( B  e.  ( fBas `  X )  /\  F  e.  ( Fil `  X
) )  ->  ( B  C_  F  <->  ( X filGen B )  C_  F
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 178    /\ wa 360    e. wcel 1726   E.wrex 2707    C_ wss 3321   ` cfv 5455  (class class class)co 6082   fBascfbas 16690   filGencfg 16691   Filcfil 17878
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 2418  ax-sep 4331  ax-nul 4339  ax-pow 4378  ax-pr 4404
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 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-nel 2603  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-sbc 3163  df-csb 3253  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-pw 3802  df-sn 3821  df-pr 3822  df-op 3824  df-uni 4017  df-br 4214  df-opab 4268  df-mpt 4269  df-id 4499  df-xp 4885  df-rel 4886  df-cnv 4887  df-co 4888  df-dm 4889  df-rn 4890  df-res 4891  df-ima 4892  df-iota 5419  df-fun 5457  df-fv 5463  df-ov 6085  df-oprab 6086  df-mpt2 6087  df-fbas 16700  df-fg 16701  df-fil 17879
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