Applying results

(i)	R4331: Probability of binary intersection with an almost sure event
(ii)	R4840: Intersection distributes over finite union
(iii)	R4841: Finite disjoint additivity of probability measure

we have \begin{equation} \begin{split} \mathbb{P}(X = x_n) & = \mathbb{P} \left( \{ X = x_n \} \cap \bigcup_{m = 1}^M \{ Y = y_m \} \right) \\ & = \mathbb{P} \left( \{ X = x_n \} \cap \bigcup_{m = 1}^M \{ Y = y_m \} \right) \\ & = \mathbb{P} \left( \bigcup_{m = 1}^M (\{ X = x_n \} \cap \{ Y = y_m \}) \right) \\ & = \mathbb{P} \left( \bigcup_{m = 1}^M (\{ X = x_n, Y = y_m \}) \right) \\ & = \sum_{m = 1}^M \mathbb{P}(X = x_n, Y = y_m) \end{split} \end{equation} $\square$