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Theorem fgfil 20221
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 20194 . . . . 5  |-  ( F  e.  ( Fil `  X
)  ->  F  e.  ( fBas `  X )
)
2 elfg 20217 . . . . 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 20199 . . . . . . . . 9  |-  ( ( F  e.  ( Fil `  X )  /\  (
x  e.  F  /\  t  C_  X  /\  x  C_  t ) )  -> 
t  e.  F )
543exp2 1214 . . . . . . . 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 2957 . . . . . 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 3515 . 2  |-  ( F  e.  ( Fil `  X
)  ->  ( X filGen F )  C_  F
)
12 ssfg 20218 . . 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 3526 1  |-  ( F  e.  ( Fil `  X
)  ->  ( X filGen F )  =  F )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   E.wrex 2818    C_ wss 3481   ` cfv 5593  (class class class)co 6294   fBascfbas 18253   filGencfg 18254   Filcfil 20191
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4573  ax-nul 4581  ax-pow 4630  ax-pr 4691
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4251  df-br 4453  df-opab 4511  df-mpt 4512  df-id 4800  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-res 5016  df-ima 5017  df-iota 5556  df-fun 5595  df-fv 5601  df-ov 6297  df-oprab 6298  df-mpt2 6299  df-fbas 18263  df-fg 18264  df-fil 20192
This theorem is referenced by:  elfilss  20222  fgtr  20236  fmid  20306  isfcf  20380  cnextcn  20412  filnetlem4  30094
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