Let $X_n$ and $Y_n$ be two sequences of random variables. For almost all realization $y$ of $Y_n$, the conditional distribution of $X_n$, given $Y_n = y$, converges to some distribution $\nu$ not depending on $y$. I want to prove that the unconditional distribution of $X_n$ also converges to $\nu$. It seems obvious because $\nu$ does not depend on $y$, but I am looking for a formal proof.
The idea behind the question relates to the conditional central limit theorem. One implication of the theorem is that the conditional distribution of some $X_n$, given $Y_n = y$, is asymptotically normal. See an example. If it is a normal distribution with mean 0 and variance 1, can I extend the result to the unconditional distribution? The distribution $\nu$ here is $N(0, 1)$ and does not depend on $y$.