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Metamath Proof Explorer

Theorem fgfil

Description: A filter generates itself. (Contributed by Jeff Hankins, 5-Sep-2009) (Revised by Stefan O'Rear, 2-Aug-2015)

Ref Expression
Assertion fgfil 
|- ( F e. ( Fil ` X ) -> ( X filGen F ) = F )

Proof

Step Hyp Ref Expression
1 filfbas 
 |-  ( F e. ( Fil ` X ) -> F e. ( fBas ` X ) )
2 elfg 
 |-  ( F e. ( fBas ` X ) -> ( t e. ( X filGen F ) <-> ( t C_ X /\ E. x e. F x C_ t ) ) )
3 1 2 syl 
 |-  ( F e. ( Fil ` X ) -> ( t e. ( X filGen F ) <-> ( t C_ X /\ E. x e. F x C_ t ) ) )
4 filss 
 |-  ( ( F e. ( Fil ` X ) /\ ( x e. F /\ t C_ X /\ x C_ t ) ) -> t e. F )
5 4 3exp2 
 |-  ( F e. ( Fil ` X ) -> ( x e. F -> ( t C_ X -> ( x C_ t -> t e. F ) ) ) )
6 5 com34 
 |-  ( F e. ( Fil ` X ) -> ( x e. F -> ( x C_ t -> ( t C_ X -> t e. F ) ) ) )
7 6 rexlimdv 
 |-  ( F e. ( Fil ` X ) -> ( E. x e. F x C_ t -> ( t C_ X -> t e. F ) ) )
8 7 impcomd 
 |-  ( F e. ( Fil ` X ) -> ( ( t C_ X /\ E. x e. F x C_ t ) -> t e. F ) )
9 3 8 sylbid 
 |-  ( F e. ( Fil ` X ) -> ( t e. ( X filGen F ) -> t e. F ) )
10 9 ssrdv 
 |-  ( F e. ( Fil ` X ) -> ( X filGen F ) C_ F )
11 ssfg 
 |-  ( F e. ( fBas ` X ) -> F C_ ( X filGen F ) )
12 1 11 syl 
 |-  ( F e. ( Fil ` X ) -> F C_ ( X filGen F ) )
13 10 12 eqssd 
 |-  ( F e. ( Fil ` X ) -> ( X filGen F ) = F )