Successor levels of the Jensen hierarchy

I prove that there is a recursive function T that does the following: Let X be transitive and rud closed, and let X ′ be the closure of X ∪{X} under rud functions. Given a Σ0 formula φ(x) and a code c for a rud function f , T (φ, c, ~x) is a Σω formula such that for any ~a ∈ X, X ′ |= φ[f(~a)] iff X |= T (φ, c, ~x)[~a]. I make this precise and show relativized versions of this. As an application, I prove that under certain conditions, if Y is the Σω extender ultrapower of X with respect to some extender F that also is an extender on X ′, then the closure of Y ∪ {Y } under rud functions is the Σ0 extender ultrapower of X ′ with respect to F , and the ultrapower embeddings agree on X.