Dense-below generic meeting lemma

If $p\in G$ and a ground-model set $D$ is dense below a forcing condition $p$, then the generic filter $G$ meets $D$. Add all incompatible forcing conditions with $p$ to $D$. The enlarged set is globally dense: a condition compatible with $p$ first has a common strengthening, then a further strengthening in $D$. Genericity meets the enlargement, and directedness excludes its incompatible part.