Users' Mathboxes Mathbox for Jeff Hankins < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  fgmin Unicode version

Theorem fgmin 26289
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 17856 . . . . . . 7  |-  ( B  e.  ( fBas `  X
)  ->  ( t  e.  ( X filGen B )  <-> 
( t  C_  X  /\  E. x  e.  B  x  C_  t ) ) )
21adantr 452 . . . . . 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 452 . . . . 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 3368 . . . . . . . . 9  |-  ( B 
C_  F  ->  ( E. x  e.  B  x  C_  t  ->  E. x  e.  F  x  C_  t
) )
54adantl 453 . . . . . . . 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 17838 . . . . . . . . . . . 12  |-  ( ( F  e.  ( Fil `  X )  /\  (
x  e.  F  /\  t  C_  X  /\  x  C_  t ) )  -> 
t  e.  F )
763exp2 1171 . . . . . . . . . . 11  |-  ( F  e.  ( Fil `  X
)  ->  ( x  e.  F  ->  ( t 
C_  X  ->  (
x  C_  t  ->  t  e.  F ) ) ) )
87com34 79 . . . . . . . . . 10  |-  ( F  e.  ( Fil `  X
)  ->  ( x  e.  F  ->  ( x 
C_  t  ->  (
t  C_  X  ->  t  e.  F ) ) ) )
98rexlimdv 2789 . . . . . . . . 9  |-  ( F  e.  ( Fil `  X
)  ->  ( E. x  e.  F  x  C_  t  ->  ( t  C_  X  ->  t  e.  F ) ) )
109ad2antlr 708 . . . . . . . 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 42 . . . . . . 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 74 . . . . . 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 421 . . . . 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 207 . . . 4  |-  ( ( ( B  e.  (
fBas `  X )  /\  F  e.  ( Fil `  X ) )  /\  B  C_  F
)  ->  ( t  e.  ( X filGen B )  ->  t  e.  F
) )
1514ssrdv 3314 . . 3  |-  ( ( ( B  e.  (
fBas `  X )  /\  F  e.  ( Fil `  X ) )  /\  B  C_  F
)  ->  ( X filGen B )  C_  F
)
1615ex 424 . 2  |-  ( ( B  e.  ( fBas `  X )  /\  F  e.  ( Fil `  X
) )  ->  ( B  C_  F  ->  ( X filGen B )  C_  F ) )
17 ssfg 17857 . . . 4  |-  ( B  e.  ( fBas `  X
)  ->  B  C_  ( X filGen B ) )
18 sstr2 3315 . . . 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 452 . 2  |-  ( ( B  e.  ( fBas `  X )  /\  F  e.  ( Fil `  X
) )  ->  (
( X filGen B ) 
C_  F  ->  B  C_  F ) )
2116, 20impbid 184 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 177    /\ wa 359    e. wcel 1721   E.wrex 2667    C_ wss 3280   ` cfv 5413  (class class class)co 6040   fBascfbas 16644   filGencfg 16645   Filcfil 17830
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-nel 2570  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-csb 3212  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-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5377  df-fun 5415  df-fv 5421  df-ov 6043  df-oprab 6044  df-mpt2 6045  df-fbas 16654  df-fg 16655  df-fil 17831
  Copyright terms: Public domain W3C validator