 Possible-values lemma for chain-condition forcing

Suppose $\mathbb{P}$ has the $\kappa$-chain condition and $\dot{f}$ is forced to be a function from an ordinal $\mu$ to an ordinal $\lambda$. For each $\xi <\mu$, a maximal antichain in a forcing order deciding $\dot{f}(\xi )$ has size below $\kappa$. Hence the ground model has a set $B_{\xi }\subseteq \lambda$ of size below $\kappa$ containing every possible value of $\dot{f}(\xi )$.