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 $\dot{f}(\xi )$.

If $\kappa$ is a regular cardinal and $\mathbb{P}$ has the $\kappa$-chain condition, then forcing with $\mathbb{P}$ preserves every cardinal number and cofinality at least $\kappa$. Indeed, a proposed surjection $f:\mu \to \lambda$ with $\mu <\lambda$ has its range contained in ${\bigcup }_{\xi <\mu }{B}_{\xi }$ from the possible-values lemma for chain-condition forcing. The regularity of $\kappa$ and infinite cardinal arithmetic make this union have cardinality below the ground-model cardinal $\lambda$, a contradiction.