Let $p$ be a set. A function $\phi$ is uniformly $\sigma_1$(p) in every admissible set if there is a $\sigma_1$ formula $\phi$ in the parameter $p$ so that $\phi$ defines $\phi$ in every $\sigma_1$-admissible set which includes $p$. A theorem of Van de Wiele states that if $\phi$ is a total function from sets to sets then $\phi$ is uniformly $\sigma_1R$ in every admissible set if anly only if it is $E$-recursive. A function is $ES_p$-recursive if it can be generated from the schemes for $E$-recursion together with a selection scheme over the transitive closure of $p$. The selection scheme is exactly what is needed to insure that the $ES_p$- recursively enumerable predicates are closed under existential quantification over the transitive closure of $p$. Two theorems are established: a) If the transitive closure of $p$ is countable than a total function on sets is $ES_p$-recursive if and only if it is uniformly $\sigma_1(p)$ in every admissible set. b) For any $p$, if $\phi$ is a function on the ordinal numbers then $\phi$ is $ES_p$-recursive if and only if it is uniformly $\sum_1(p)$ in every admissible set.