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Theorem fgfil 17586
Description: A filter generates itself. (Contributed by Jeff Hankins, 5-Sep-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)
Assertion
Ref Expression
fgfil  |-  ( F  e.  ( Fil `  X
)  ->  ( X filGen F )  =  F )

Proof of Theorem fgfil
Dummy variables  x  t are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 filfbas 17559 . . . . 5  |-  ( F  e.  ( Fil `  X
)  ->  F  e.  ( fBas `  X )
)
2 elfg 17582 . . . . 5  |-  ( F  e.  ( fBas `  X
)  ->  ( t  e.  ( X filGen F )  <-> 
( t  C_  X  /\  E. x  e.  F  x  C_  t ) ) )
31, 2syl 15 . . . 4  |-  ( F  e.  ( Fil `  X
)  ->  ( t  e.  ( X filGen F )  <-> 
( t  C_  X  /\  E. x  e.  F  x  C_  t ) ) )
4 filss 17564 . . . . . . . . 9  |-  ( ( F  e.  ( Fil `  X )  /\  (
x  e.  F  /\  t  C_  X  /\  x  C_  t ) )  -> 
t  e.  F )
543exp2 1169 . . . . . . . 8  |-  ( F  e.  ( Fil `  X
)  ->  ( x  e.  F  ->  ( t 
C_  X  ->  (
x  C_  t  ->  t  e.  F ) ) ) )
65com34 77 . . . . . . 7  |-  ( F  e.  ( Fil `  X
)  ->  ( x  e.  F  ->  ( x 
C_  t  ->  (
t  C_  X  ->  t  e.  F ) ) ) )
76rexlimdv 2679 . . . . . 6  |-  ( F  e.  ( Fil `  X
)  ->  ( E. x  e.  F  x  C_  t  ->  ( t  C_  X  ->  t  e.  F ) ) )
87com23 72 . . . . 5  |-  ( F  e.  ( Fil `  X
)  ->  ( t  C_  X  ->  ( E. x  e.  F  x  C_  t  ->  t  e.  F ) ) )
98imp3a 420 . . . 4  |-  ( F  e.  ( Fil `  X
)  ->  ( (
t  C_  X  /\  E. x  e.  F  x 
C_  t )  -> 
t  e.  F ) )
103, 9sylbid 206 . . 3  |-  ( F  e.  ( Fil `  X
)  ->  ( t  e.  ( X filGen F )  ->  t  e.  F
) )
1110ssrdv 3198 . 2  |-  ( F  e.  ( Fil `  X
)  ->  ( X filGen F )  C_  F
)
12 ssfg 17583 . . 3  |-  ( F  e.  ( fBas `  X
)  ->  F  C_  ( X filGen F ) )
131, 12syl 15 . 2  |-  ( F  e.  ( Fil `  X
)  ->  F  C_  ( X filGen F ) )
1411, 13eqssd 3209 1  |-  ( F  e.  ( Fil `  X
)  ->  ( X filGen F )  =  F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1632    e. wcel 1696   E.wrex 2557    C_ wss 3165   ` cfv 5271  (class class class)co 5874   fBascfbas 17534   filGencfg 17535   Filcfil 17556
This theorem is referenced by:  elfilss  17587  fgtr  17601  fmid  17671  isfcf  17745  fil2ss  25660  filnetlem4  26433
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-nel 2462  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-csb 3095  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fv 5279  df-ov 5877  df-oprab 5878  df-mpt2 5879  df-fbas 17536  df-fg 17537  df-fil 17557
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