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Theorem fgmin 30811
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 20817 . . . . . . 7  |-  ( B  e.  ( fBas `  X
)  ->  ( t  e.  ( X filGen B )  <-> 
( t  C_  X  /\  E. x  e.  B  x  C_  t ) ) )
21adantr 466 . . . . . 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 466 . . . . 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 3532 . . . . . . . . 9  |-  ( B 
C_  F  ->  ( E. x  e.  B  x  C_  t  ->  E. x  e.  F  x  C_  t
) )
54adantl 467 . . . . . . . 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 20799 . . . . . . . . . . . 12  |-  ( ( F  e.  ( Fil `  X )  /\  (
x  e.  F  /\  t  C_  X  /\  x  C_  t ) )  -> 
t  e.  F )
763exp2 1223 . . . . . . . . . . 11  |-  ( F  e.  ( Fil `  X
)  ->  ( x  e.  F  ->  ( t 
C_  X  ->  (
x  C_  t  ->  t  e.  F ) ) ) )
87com34 86 . . . . . . . . . 10  |-  ( F  e.  ( Fil `  X
)  ->  ( x  e.  F  ->  ( x 
C_  t  ->  (
t  C_  X  ->  t  e.  F ) ) ) )
98rexlimdv 2922 . . . . . . . . 9  |-  ( F  e.  ( Fil `  X
)  ->  ( E. x  e.  F  x  C_  t  ->  ( t  C_  X  ->  t  e.  F ) ) )
109ad2antlr 731 . . . . . . . 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 45 . . . . . . 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 81 . . . . . 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 ) ) )
1312impd 432 . . . . 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 218 . . . 4  |-  ( ( ( B  e.  (
fBas `  X )  /\  F  e.  ( Fil `  X ) )  /\  B  C_  F
)  ->  ( t  e.  ( X filGen B )  ->  t  e.  F
) )
1514ssrdv 3476 . . 3  |-  ( ( ( B  e.  (
fBas `  X )  /\  F  e.  ( Fil `  X ) )  /\  B  C_  F
)  ->  ( X filGen B )  C_  F
)
1615ex 435 . 2  |-  ( ( B  e.  ( fBas `  X )  /\  F  e.  ( Fil `  X
) )  ->  ( B  C_  F  ->  ( X filGen B )  C_  F ) )
17 ssfg 20818 . . . 4  |-  ( B  e.  ( fBas `  X
)  ->  B  C_  ( X filGen B ) )
18 sstr2 3477 . . . 4  |-  ( B 
C_  ( X filGen B )  ->  ( ( X filGen B )  C_  F  ->  B  C_  F
) )
1917, 18syl 17 . . 3  |-  ( B  e.  ( fBas `  X
)  ->  ( ( X filGen B )  C_  F  ->  B  C_  F
) )
2019adantr 466 . 2  |-  ( ( B  e.  ( fBas `  X )  /\  F  e.  ( Fil `  X
) )  ->  (
( X filGen B ) 
C_  F  ->  B  C_  F ) )
2116, 20impbid 193 1  |-  ( ( B  e.  ( fBas `  X )  /\  F  e.  ( Fil `  X
) )  ->  ( B  C_  F  <->  ( X filGen B )  C_  F
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    e. wcel 1870   E.wrex 2783    C_ wss 3442   ` cfv 5601  (class class class)co 6305   fBascfbas 18893   filGencfg 18894   Filcfil 20791
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-nel 2628  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-ima 4867  df-iota 5565  df-fun 5603  df-fv 5609  df-ov 6308  df-oprab 6309  df-mpt2 6310  df-fbas 18902  df-fg 18903  df-fil 20792
This theorem is referenced by: (None)
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