Theorem locfinbas

Description: A locally finite cover must cover the base set of its corresponding topological space. (Contributed by Jeff Hankins, 21-Jan-2010)

	Ref	Expression
Hypotheses	locfinbas.1	|- X = U. J
	locfinbas.2	|- Y = U. A
Assertion	locfinbas	|- ( A e. ( LocFin ` J ) -> X = Y )

Proof

Step	Hyp	Ref	Expression
1		locfinbas.1	|- X = U. J
2		locfinbas.2	|- Y = U. A
3	1 2	islocfin	|- ( A e. ( LocFin ` J ) <-> ( J e. Top /\ X = Y /\ A. s e. X E. n e. J ( s e. n /\ { x e. A | ( x i^i n ) =/= (/) } e. Fin ) ) )
4	3	simp2bi	|- ( A e. ( LocFin ` J ) -> X = Y )