On the equation image

This note uses some recent powerful tools related with semidirect products V*W of pseudovarieties of semigroups, particularly when the second factor is the pseudovariety G of all finite groups, to give syntactic proofs of equalities of the form image, where image denotes the pseudovariety consisting of the finite semigroups whose idempotents generate subsemigroups from V. Subpseudovarieties V of image are considered, including all subpseudovarieties of image itself, and J, the latter providing a new proof of a result of Henckell and Rhodes.