Expressing the probability of a random vector being in a random set as a conditional expectation

Let $X$ be a random vector in $\mathbb{R}^n$ and let $\mathbb{P}_X$ be its distribution. Let $\{A_y : y \in \mathbb{R}^m\}$ be some collection of Borel subsets of $\mathbb{R}^n$ and let $Y$ be a random vector in $\mathbb{R}^m$. Is it true that $\mathbb{P}_X(A_Y) = \mathbb{E}[\mathbb{I}_{A_Y}(X) \mid Y]$, where $\mathbb{I}_{A_y}$ is the indicator function of $A_y$? It seems reasonable to me, but I don't see how to prove this.