Metamath Proof Explorer


Theorem fibas

Description: A collection of finite intersections is a basis. The initial set is a subbasis for the topology. (Contributed by Jeff Hankins, 25-Aug-2009) (Revised by Mario Carneiro, 24-Nov-2013)

Ref Expression
Assertion fibas
|- ( fi ` A ) e. TopBases

Proof

Step Hyp Ref Expression
1 fvex
 |-  ( fi ` A ) e. _V
2 fiin
 |-  ( ( x e. ( fi ` A ) /\ y e. ( fi ` A ) ) -> ( x i^i y ) e. ( fi ` A ) )
3 2 rgen2
 |-  A. x e. ( fi ` A ) A. y e. ( fi ` A ) ( x i^i y ) e. ( fi ` A )
4 fiinbas
 |-  ( ( ( fi ` A ) e. _V /\ A. x e. ( fi ` A ) A. y e. ( fi ` A ) ( x i^i y ) e. ( fi ` A ) ) -> ( fi ` A ) e. TopBases )
5 1 3 4 mp2an
 |-  ( fi ` A ) e. TopBases