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Theorem fgfil 19590
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 19563 . . . . 5  |-  ( F  e.  ( Fil `  X
)  ->  F  e.  ( fBas `  X )
)
2 elfg 19586 . . . . 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 19568 . . . . . . . . 9  |-  ( ( F  e.  ( Fil `  X )  /\  (
x  e.  F  /\  t  C_  X  /\  x  C_  t ) )  -> 
t  e.  F )
543exp2 1206 . . . . . . . 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 2946 . . . . . 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 3473 . 2  |-  ( F  e.  ( Fil `  X
)  ->  ( X filGen F )  C_  F
)
12 ssfg 19587 . . 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 3484 1  |-  ( F  e.  ( Fil `  X
)  ->  ( X filGen F )  =  F )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1370    e. wcel 1758   E.wrex 2800    C_ wss 3439   ` cfv 5529  (class class class)co 6203   fBascfbas 17939   filGencfg 17940   Filcfil 19560
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-nel 2651  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-ima 4964  df-iota 5492  df-fun 5531  df-fv 5537  df-ov 6206  df-oprab 6207  df-mpt2 6208  df-fbas 17949  df-fg 17950  df-fil 19561
This theorem is referenced by:  elfilss  19591  fgtr  19605  fmid  19675  isfcf  19749  cnextcn  19781  filnetlem4  28773
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