"Show that fY|X(y) is a density function of a true probability distribution"

Where do I begin?? The course I'm taking is focusing heavily on proofs like this, and I'm studying it at a distance so my opportunity to ask questions are very limited. I've ordered a book dealing with how to read and do mathematical proofs in general, but I would love to get some input from someone specifically on proofs relating to probability such as these (and if there are other resources).

How do I go about showing that? I understand the idea of integrating to get the probability mass under the curve and that it has to = 1 as x goes from -infinity to infinity, but how do I prove that this does that?

I'm just not sure where to start...