Notion of conditional weak convergence

I am looking for references or lecture notes which define the notion of conditional weak convergence of a sequence of random variables. In the case of (usual) weak convergence, we say that a sequence $(X_n)_{n\geq1}$ of random variables weakly converges to $X$ as $n\rightarrow \infty$ if for all bounded and continuous functions $f$, we have the convergence of $\mathbb{E}(f(X_n))$ to $\mathbb{E}(f(X))$ as $n\rightarrow \infty$. My understanding of conditional weak convergence would be the following definition:

Let $(Y_n)_{n\geq1}$ be a sequence of random variables which weakly converges to $Y$. We say that a sequence $(X_n)_{n\geq1}$ of random variables weakly converges to $X$ conditionally on $(Y_n)_{n\geq1}$ as $n\rightarrow \infty$ if for all bounded and continuous functions $f$ and any Borel set $A$, we have the convergence of $\mathbb{E}(f(X_n)|Y_n\in A)$ to $\mathbb{E}(f(X)|Y\in A)$ as $n\rightarrow \infty$.

I am wondering whether this is the correct way of defining (or characterising) conditional weak convergence? Any comments or references to literature would be greatly appreciated.