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Theorem fgmin 25485
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
StepHypRef Expression
1 elfg 17398 . . . . . . 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 3159 . . . . . . . . 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 17380 . . . . . . . . . . . 12  |-  ( ( F  e.  ( Fil `  X )  /\  (
x  e.  F  /\  t  C_  X  /\  x  C_  t ) )  -> 
t  e.  F )
763exp2 1174 . . . . . . . . . . 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 2628 . . . . . . . . 9  |-  ( F  e.  ( Fil `  X
)  ->  ( E. x  e.  F  x  C_  t  ->  ( t  C_  X  ->  t  e.  F ) ) )
109ad2antlr 710 . . . . . . . 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 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 3106 . . 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 17399 . . . 4  |-  ( B  e.  ( fBas `  X
)  ->  B  C_  ( X filGen B ) )
18 sstr2 3107 . . . 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 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 6    <-> wb 178    /\ wa 360    e. wcel 1621   E.wrex 2510    C_ wss 3078   ` cfv 4592  (class class class)co 5710   fBascfbas 17350   filGencfg 17351   Filcfil 17372
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-nel 2415  df-ral 2513  df-rex 2514  df-rab 2516  df-v 2729  df-sbc 2922  df-csb 3010  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-op 3553  df-uni 3728  df-br 3921  df-opab 3975  df-mpt 3976  df-id 4202  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fv 4608  df-ov 5713  df-oprab 5714  df-mpt2 5715  df-fbas 17352  df-fg 17353  df-fil 17373
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