∈-representation and set-prolongations

By an ∈-representation of a relation we mean its isomorphic embedding to E = {〈x, y〉; x ∈ y}. Some theorems on such a representation are presented. Especially, we prove a version of the well-known theorem on isomorphic representation of extensional and well-founded relations in E, which holds in Zermelo-Fraenkel set theory. This our version is in Zermelo-Fraenkel set theory false. A general theorem on a set-prolongation is proved; it enables us to solve the task of the representation in question.