If $\mathbb P_{Y|X\in A}(B):=\mathbb P(Y\in B \ | \ X \in A)$ can we explicitly define the r.v. $(Y|X\in A)$?

Question

When introducing conditional expectation, one can define $\mathbb P_{Y|X\in A}(B):=\mathbb P(Y\in B \ | \ X \in A)$ which is itself a law. I was wondering if there is a way to define a random variable which has it as its law. So

\begin{align*} (Y|X\in A): \Omega & \to \mathbb R \\ \omega & \mapsto ? \\ & \end{align*}