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Metamath Proof Explorer

Theorem fgss

Description: A bigger base generates a bigger filter. (Contributed by NM, 5-Sep-2009) (Revised by Stefan O'Rear, 2-Aug-2015)

Ref Expression
Assertion fgss 
|- ( ( F e. ( fBas ` X ) /\ G e. ( fBas ` X ) /\ F C_ G ) -> ( X filGen F ) C_ ( X filGen G ) )

Proof

Step Hyp Ref Expression
1 ssrexv 
 |-  ( F C_ G -> ( E. x e. F x C_ t -> E. x e. G x C_ t ) )
2 1 anim2d 
 |-  ( F C_ G -> ( ( t C_ X /\ E. x e. F x C_ t ) -> ( t C_ X /\ E. x e. G x C_ t ) ) )
3 2 3ad2ant3 
 |-  ( ( F e. ( fBas ` X ) /\ G e. ( fBas ` X ) /\ F C_ G ) -> ( ( t C_ X /\ E. x e. F x C_ t ) -> ( t C_ X /\ E. x e. G x C_ t ) ) )
4 elfg 
 |-  ( F e. ( fBas ` X ) -> ( t e. ( X filGen F ) <-> ( t C_ X /\ E. x e. F x C_ t ) ) )
5 4 3ad2ant1 
 |-  ( ( F e. ( fBas ` X ) /\ G e. ( fBas ` X ) /\ F C_ G ) -> ( t e. ( X filGen F ) <-> ( t C_ X /\ E. x e. F x C_ t ) ) )
6 elfg 
 |-  ( G e. ( fBas ` X ) -> ( t e. ( X filGen G ) <-> ( t C_ X /\ E. x e. G x C_ t ) ) )
7 6 3ad2ant2 
 |-  ( ( F e. ( fBas ` X ) /\ G e. ( fBas ` X ) /\ F C_ G ) -> ( t e. ( X filGen G ) <-> ( t C_ X /\ E. x e. G x C_ t ) ) )
8 3 5 7 3imtr4d 
 |-  ( ( F e. ( fBas ` X ) /\ G e. ( fBas ` X ) /\ F C_ G ) -> ( t e. ( X filGen F ) -> t e. ( X filGen G ) ) )
9 8 ssrdv 
 |-  ( ( F e. ( fBas ` X ) /\ G e. ( fBas ` X ) /\ F C_ G ) -> ( X filGen F ) C_ ( X filGen G ) )