Theorem bastop

Description: Two ways to express that a basis is a topology. (Contributed by NM, 18-Jul-2006)

	Ref	Expression
Assertion	bastop	|- ( B e. TopBases -> ( B e. Top <-> ( topGen ` B ) = B ) )

Proof

Step	Hyp	Ref	Expression
1		tgtop	|- ( B e. Top -> ( topGen ` B ) = B )
2		tgcl	|- ( B e. TopBases -> ( topGen ` B ) e. Top )
3		eleq1	|- ( ( topGen ` B ) = B -> ( ( topGen ` B ) e. Top <-> B e. Top ) )
4	2 3	syl5ibcom	|- ( B e. TopBases -> ( ( topGen ` B ) = B -> B e. Top ) )
5	1 4	impbid2	|- ( B e. TopBases -> ( B e. Top <-> ( topGen ` B ) = B ) )