A generalization of Solovay's $\mathrm{\Sigma }$-construction

A $\mathrm{\Sigma }$-construction of Solovay is partially extended to the case of intermediate sets which are not necessarily subsets of the ground model. As an application, we prove that, for a given name $t$, the set of all sets $t[G]$, $G$ being generic over the ground model, is Borel. This result was first established by Zapletal by a totally different descriptive set theoretic argument.