Metamath Proof Explorer

Theorem tg1

Description: Property of a member of a topology generated by a basis. (Contributed by NM, 20-Jul-2006)

Ref Expression
Assertion tg1 
|- ( A e. ( topGen ` B ) -> A C_ U. B )

Proof

Step Hyp Ref Expression
1 elfvdm 
 |-  ( A e. ( topGen ` B ) -> B e. dom topGen )
2 eltg2 
 |-  ( B e. dom topGen -> ( A e. ( topGen ` B ) <-> ( A C_ U. B /\ A. x e. A E. y e. B ( x e. y /\ y C_ A ) ) ) )
3 2 simprbda 
 |-  ( ( B e. dom topGen /\ A e. ( topGen ` B ) ) -> A C_ U. B )
4 1 3 mpancom 
 |-  ( A e. ( topGen ` B ) -> A C_ U. B )