I'm trying to define a simple conditional PMF,
$$ \begin{equation} p_{X\mid Y}(x\mid y) = \Pr(X = x \mid Y = y) = \frac{\Pr(X=x\, \cap\, Y=y)}{\Pr(Y = y)}. \end{equation} $$
The problem is that sometimes I only know a distribution for $Y$ so something like
$$p_{X\mid Y\sim D}(x\mid d) = \Pr(X = x \mid Y\sim d).$$
(I made up the above notation to convey my idea. I realize it is probably completely wrong)
Is there a simple way to express this idea in formal writing? I'd like to just put $d$ in for $y$ but $\Pr(X = x \mid Y = d)$ seems like an abuse of notation. Do I have to define two separate PMF's? Also, extending my outcome space $\Omega$ to include distributions $D$ over $Y$ seems wrong too, so I don't want to make $d$ an event...