It's literally the transitive closure of ∈↾x. By definition, ∈↾S is transitive iff ∀x,y,z∈S ( x∈y∈z ⇒ x∈z ). So we want the minimal set S ⊇ x such that ∈↾S is transitive. And the construction you wrote down is exactly that, and it is easy to prove it.