Let $(\Omega, \mathcal{A}, \mathbb{P})$ be a probability space and let $X$ and $Y$ be random variables defined on that space, mapping to $(\mathcal{X}, \mathcal{C})$ and $(\mathcal{Y}, \mathcal{D})$, respectively.
The distribution of $X$ is the pushforward measure $\mathbb{P}_X(C):=\mathbb{P}(X\in C):=\mathbb{P}(\{\omega\in\Omega:X(\omega)=C\})$, for all $C\in\mathcal{C}$.
My question is now how the conditional distribution can be written in terms of a set $\{\omega\in\Omega: ?\}$, i.e., $\mathbb{P}_{X\mid Y=y}(C)=\mathbb{P}(X\in C \mid Y=y)=\mathbb{P}(\{\omega\in\Omega:?\})$
So if someone comes across this question: It really just means
$\mathbb{P}(X\in C|Y)=\mathbb{P}(X^{-1}(C)|Y)=\mathbb{P}(\{\omega\in\Omega:X(\omega)=C\}|Y)$
See Definition 26 in https://www.stat.cmu.edu/~arinaldo/Teaching/36752/S18/Notes/lec_notes_6.pdf
where conditioning on $Y$ means conditioning on the sigma algebra generated by $Y$.