The matching problem defines a transferable utility game. A payoff is in the core of the matching game if no subset $S$ of individuals can improve themselves by breaking away.

Theorem. Any stable payoff is a core payoff.

Proof. Consider without loss of generality the coalition containing workers 1 through $k$ and firms 1 through $k$. Suppose that the optimal matching in the coalition matches each worker $i$ with firm $i$. For any stable payoff $(w^*,\pi^*)$ for the entire group, $w^*_i+\pi^*_i\geq v_{ii}$, so \begin{equation*} \sum_{i=1}^kw^*_i+\pi^*_i\geq \sum_{i=1}^kv_{ii}=v_P(S). \end{equation*} Thus coalition $S$ cannot improve upon any stable payoff.

The converse is obvious. ∎