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Theorem fgfil 20242
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 20215 . . . . 5  |-  ( F  e.  ( Fil `  X
)  ->  F  e.  ( fBas `  X )
)
2 elfg 20238 . . . . 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 20220 . . . . . . . . 9  |-  ( ( F  e.  ( Fil `  X )  /\  (
x  e.  F  /\  t  C_  X  /\  x  C_  t ) )  -> 
t  e.  F )
543exp2 1213 . . . . . . . 8  |-  ( F  e.  ( Fil `  X
)  ->  ( x  e.  F  ->  ( t 
C_  X  ->  (
x  C_  t  ->  t  e.  F ) ) ) )
65com34 83 . . . . . . 7  |-  ( F  e.  ( Fil `  X
)  ->  ( x  e.  F  ->  ( x 
C_  t  ->  (
t  C_  X  ->  t  e.  F ) ) ) )
76rexlimdv 2931 . . . . . 6  |-  ( F  e.  ( Fil `  X
)  ->  ( E. x  e.  F  x  C_  t  ->  ( t  C_  X  ->  t  e.  F ) ) )
87com23 78 . . . . 5  |-  ( F  e.  ( Fil `  X
)  ->  ( t  C_  X  ->  ( E. x  e.  F  x  C_  t  ->  t  e.  F ) ) )
98impd 431 . . . 4  |-  ( F  e.  ( Fil `  X
)  ->  ( (
t  C_  X  /\  E. x  e.  F  x 
C_  t )  -> 
t  e.  F ) )
103, 9sylbid 215 . . 3  |-  ( F  e.  ( Fil `  X
)  ->  ( t  e.  ( X filGen F )  ->  t  e.  F
) )
1110ssrdv 3492 . 2  |-  ( F  e.  ( Fil `  X
)  ->  ( X filGen F )  C_  F
)
12 ssfg 20239 . . 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 3503 1  |-  ( F  e.  ( Fil `  X
)  ->  ( X filGen F )  =  F )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1381    e. wcel 1802   E.wrex 2792    C_ wss 3458   ` cfv 5574  (class class class)co 6277   fBascfbas 18274   filGencfg 18275   Filcfil 20212
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-nel 2639  df-ral 2796  df-rex 2797  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-op 4017  df-uni 4231  df-br 4434  df-opab 4492  df-mpt 4493  df-id 4781  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fv 5582  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-fbas 18284  df-fg 18285  df-fil 20213
This theorem is referenced by:  elfilss  20243  fgtr  20257  fmid  20327  isfcf  20401  cnextcn  20433  filnetlem4  30167
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