Possible-values lemma for chain-condition forcing
= 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 $\dot f(\xi)$.