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Theorem fgfil 17899
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 17872 . . . . 5  |-  ( F  e.  ( Fil `  X
)  ->  F  e.  ( fBas `  X )
)
2 elfg 17895 . . . . 5  |-  ( F  e.  ( fBas `  X
)  ->  ( t  e.  ( X filGen F )  <-> 
( t  C_  X  /\  E. x  e.  F  x  C_  t ) ) )
31, 2syl 16 . . . 4  |-  ( F  e.  ( Fil `  X
)  ->  ( t  e.  ( X filGen F )  <-> 
( t  C_  X  /\  E. x  e.  F  x  C_  t ) ) )
4 filss 17877 . . . . . . . . 9  |-  ( ( F  e.  ( Fil `  X )  /\  (
x  e.  F  /\  t  C_  X  /\  x  C_  t ) )  -> 
t  e.  F )
543exp2 1171 . . . . . . . 8  |-  ( F  e.  ( Fil `  X
)  ->  ( x  e.  F  ->  ( t 
C_  X  ->  (
x  C_  t  ->  t  e.  F ) ) ) )
65com34 79 . . . . . . 7  |-  ( F  e.  ( Fil `  X
)  ->  ( x  e.  F  ->  ( x 
C_  t  ->  (
t  C_  X  ->  t  e.  F ) ) ) )
76rexlimdv 2821 . . . . . 6  |-  ( F  e.  ( Fil `  X
)  ->  ( E. x  e.  F  x  C_  t  ->  ( t  C_  X  ->  t  e.  F ) ) )
87com23 74 . . . . 5  |-  ( F  e.  ( Fil `  X
)  ->  ( t  C_  X  ->  ( E. x  e.  F  x  C_  t  ->  t  e.  F ) ) )
98imp3a 421 . . . 4  |-  ( F  e.  ( Fil `  X
)  ->  ( (
t  C_  X  /\  E. x  e.  F  x 
C_  t )  -> 
t  e.  F ) )
103, 9sylbid 207 . . 3  |-  ( F  e.  ( Fil `  X
)  ->  ( t  e.  ( X filGen F )  ->  t  e.  F
) )
1110ssrdv 3346 . 2  |-  ( F  e.  ( Fil `  X
)  ->  ( X filGen F )  C_  F
)
12 ssfg 17896 . . 3  |-  ( F  e.  ( fBas `  X
)  ->  F  C_  ( X filGen F ) )
131, 12syl 16 . 2  |-  ( F  e.  ( Fil `  X
)  ->  F  C_  ( X filGen F ) )
1411, 13eqssd 3357 1  |-  ( F  e.  ( Fil `  X
)  ->  ( X filGen F )  =  F )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   E.wrex 2698    C_ wss 3312   ` cfv 5446  (class class class)co 6073   fBascfbas 16681   filGencfg 16682   Filcfil 17869
This theorem is referenced by:  elfilss  17900  fgtr  17914  fmid  17984  isfcf  18058  cnextcn  18090  filnetlem4  26401
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pow 4369  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-nel 2601  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-csb 3244  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-mpt 4260  df-id 4490  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-rn 4881  df-res 4882  df-ima 4883  df-iota 5410  df-fun 5448  df-fv 5454  df-ov 6076  df-oprab 6077  df-mpt2 6078  df-fbas 16691  df-fg 16692  df-fil 17870
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