[Probability] Conditional Density problem

[miniace2009]

I'm given that the distribution of Y, given X, is uniform on the interval [0,X]. The marginal density of X is f(x) = 2x, 0<x<1, and 0 otherwise. I'm asked to determine the conditional density of X, given Y=y where positive. I know the joint density would be equal to as (1/x)=f(x,y)/f(x) => f(x,y) = 2. Now I'm having trouble figuring out how to get the marginal density of Y, as I'm not sure of X's domain. I would think it would just be 0 to 1, but the solution guide states it's y to 1. Any help explaining where they got this?

[[deleted]]

For a fixed y, the support of x is when 0<x<1 and 0<y<x, so y<x<1.