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Metamath Proof Explorer

Theorem fbsspw

Description: A filter base on a set is a subset of the power set. (Contributed by Stefan O'Rear, 28-Jul-2015)

Ref Expression
Assertion fbsspw 
|- ( F e. ( fBas ` B ) -> F C_ ~P B )

Proof

Step Hyp Ref Expression
1 elfvdm 
 |-  ( F e. ( fBas ` B ) -> B e. dom fBas )
2 isfbas 
 |-  ( B e. dom fBas -> ( F e. ( fBas ` B ) <-> ( F C_ ~P B /\ ( F =/= (/) /\ (/) e/ F /\ A. x e. F A. y e. F ( F i^i ~P ( x i^i y ) ) =/= (/) ) ) ) )
3 1 2 syl 
 |-  ( F e. ( fBas ` B ) -> ( F e. ( fBas ` B ) <-> ( F C_ ~P B /\ ( F =/= (/) /\ (/) e/ F /\ A. x e. F A. y e. F ( F i^i ~P ( x i^i y ) ) =/= (/) ) ) ) )
4 3 ibi 
 |-  ( F e. ( fBas ` B ) -> ( F C_ ~P B /\ ( F =/= (/) /\ (/) e/ F /\ A. x e. F A. y e. F ( F i^i ~P ( x i^i y ) ) =/= (/) ) ) )
5 4 simpld 
 |-  ( F e. ( fBas ` B ) -> F C_ ~P B )