I am reading a paper that discusses mathcing of workers to firms in the labor market (see the excerpt below). Here an outcome $(\mu,\pi,w)$ consists of a matching function pairing workers to firms, a profit vector for firms, and a wage vector for workers. The text states that the outcome is stable if there is no blocking pair and that this happens if $w(x) + \pi(y) >= f(x,y) \forall (x,y)$. My question is: how does this condition ensure a stable outcome?
I understand that this condition requires that the sum of the wage and profit in the existing pair be larger than any pair. And I understand that this means that a firm can always increase wages to keep their preferred worker from deviating to another firm. But that implies that the matching function mu is stable. I don't understand what in the above condition states that the wage and profit vectors in the stable outcome are actually selected to be stable?