Let $\mathrm{KP}^-$ be the theory resulting from Kripke-Platek set theory by restricting Foundation to Set Foundation. Let $G: V \rightarrow V (V:=$ universe of sets) be a $\triangle_0$-definable set function, i.e. there is a $\triangle_0$-formula $\varphi(x, y)$ such that $\varphi(x, G(x))$ is true for all sets $x$, and $V \models \forall x \exists!y\varphi (x, y)$. In this paper we shall verify (by elementary proof-theoretic methods) that the collection of set functions primitive recursive in $G$ coincides with the collection of those functions which are $\Sigma_1$-definable in $\mathrm{KP}^- + \Sigma_1$-Foundation $+ \forall x \exists!y\varphi (x, y)$. Moreover, we show that this is still true if one adds $\Pi_1$-Foundation or a weak version of $\triangle_0$-Dependent Choices to the latter theory.