We've just started continuous probability and I'm having a bit of trouble getting my head around this one.

A continuous random variable A takes values in (p,q) with c.d.f. F, while we know that F is strictly increasing on (p,q). I want to show that Y=F(X) is uniformly distributed on (0,1). However, I'm not quite sure how to proceed. Can we use the formula relating the c.d.f. and the p.d.f. of X? I don't understand how F can be 'applied to' X in this context.