Metamath Proof Explorer

Theorem ssfg

Description: A filter base is a subset of its generated filter. (Contributed by Jeff Hankins, 3-Sep-2009) (Revised by Stefan O'Rear, 2-Aug-2015)

Ref Expression
Assertion ssfg 
|- ( F e. ( fBas ` X ) -> F C_ ( X filGen F ) )

Proof

Step Hyp Ref Expression
1 fbelss 
 |-  ( ( F e. ( fBas ` X ) /\ t e. F ) -> t C_ X )
2 1 ex 
 |-  ( F e. ( fBas ` X ) -> ( t e. F -> t C_ X ) )
3 ssid 
 |-  t C_ t
4 sseq1 
 |-  ( x = t -> ( x C_ t <-> t C_ t ) )
5 4 rspcev 
 |-  ( ( t e. F /\ t C_ t ) -> E. x e. F x C_ t )
6 3 5 mpan2 
 |-  ( t e. F -> E. x e. F x C_ t )
7 2 6 jca2 
 |-  ( F e. ( fBas ` X ) -> ( t e. F -> ( t C_ X /\ E. x e. F x C_ t ) ) )
8 elfg 
 |-  ( F e. ( fBas ` X ) -> ( t e. ( X filGen F ) <-> ( t C_ X /\ E. x e. F x C_ t ) ) )
9 7 8 sylibrd 
 |-  ( F e. ( fBas ` X ) -> ( t e. F -> t e. ( X filGen F ) ) )
10 9 ssrdv 
 |-  ( F e. ( fBas ` X ) -> F C_ ( X filGen F ) )