Is the conditional probability uniform if the joint is uniform?

I mean this questions in the general sense. Suppose, we have two random variables $X$ and $Y$ that are jointly uniformly distributed, i.e.
$$f_{X,Y}(x,y) = k space forall x,y$$

This should mean that all (x,y) points are equally likely. Now if we find out one of the X’s or Y’s, we’re left with just finding the other one using the conditional distribution. So,
$$f_{X|Y}(x|y) = frac{f_{X,Y}(x,y)}{f_Y(y)} = frac{k}{f_Y(y)}$$

So given we know $Y = y$, the joint uniform density then becomes a conditional uniform density? Is this correct? It should also hold for the discrete case. Can someone point me to a more mathematical proof?