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Theorem fgmin 29819
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 20135 . . . . . . 7  |-  ( B  e.  ( fBas `  X
)  ->  ( t  e.  ( X filGen B )  <-> 
( t  C_  X  /\  E. x  e.  B  x  C_  t ) ) )
21adantr 465 . . . . . 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 465 . . . . 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 3565 . . . . . . . . 9  |-  ( B 
C_  F  ->  ( E. x  e.  B  x  C_  t  ->  E. x  e.  F  x  C_  t
) )
54adantl 466 . . . . . . . 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 20117 . . . . . . . . . . . 12  |-  ( ( F  e.  ( Fil `  X )  /\  (
x  e.  F  /\  t  C_  X  /\  x  C_  t ) )  -> 
t  e.  F )
763exp2 1214 . . . . . . . . . . 11  |-  ( F  e.  ( Fil `  X
)  ->  ( x  e.  F  ->  ( t 
C_  X  ->  (
x  C_  t  ->  t  e.  F ) ) ) )
87com34 83 . . . . . . . . . 10  |-  ( F  e.  ( Fil `  X
)  ->  ( x  e.  F  ->  ( x 
C_  t  ->  (
t  C_  X  ->  t  e.  F ) ) ) )
98rexlimdv 2953 . . . . . . . . 9  |-  ( F  e.  ( Fil `  X
)  ->  ( E. x  e.  F  x  C_  t  ->  ( t  C_  X  ->  t  e.  F ) ) )
109ad2antlr 726 . . . . . . . 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 44 . . . . . . 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 78 . . . . . 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 431 . . . . 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 215 . . . 4  |-  ( ( ( B  e.  (
fBas `  X )  /\  F  e.  ( Fil `  X ) )  /\  B  C_  F
)  ->  ( t  e.  ( X filGen B )  ->  t  e.  F
) )
1514ssrdv 3510 . . 3  |-  ( ( ( B  e.  (
fBas `  X )  /\  F  e.  ( Fil `  X ) )  /\  B  C_  F
)  ->  ( X filGen B )  C_  F
)
1615ex 434 . 2  |-  ( ( B  e.  ( fBas `  X )  /\  F  e.  ( Fil `  X
) )  ->  ( B  C_  F  ->  ( X filGen B )  C_  F ) )
17 ssfg 20136 . . . 4  |-  ( B  e.  ( fBas `  X
)  ->  B  C_  ( X filGen B ) )
18 sstr2 3511 . . . 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 465 . 2  |-  ( ( B  e.  ( fBas `  X )  /\  F  e.  ( Fil `  X
) )  ->  (
( X filGen B ) 
C_  F  ->  B  C_  F ) )
2116, 20impbid 191 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 184    /\ wa 369    e. wcel 1767   E.wrex 2815    C_ wss 3476   ` cfv 5588  (class class class)co 6284   fBascfbas 18205   filGencfg 18206   Filcfil 20109
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fv 5596  df-ov 6287  df-oprab 6288  df-mpt2 6289  df-fbas 18215  df-fg 18216  df-fil 20110
This theorem is referenced by: (None)
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