Let $\varphi:\mathbb{N}\to\mathbb{N}^*$ be an arbitrary recursive enumeration of finite strings and $\mathcal{I}^n_i(x_1,...,x_n) = x_i $ be the $i$-th projection over $n$ variables.
I would like to show from the point of view of recursion theory that the "variable" projection $$ (n,y) \mapsto \mathcal{I}_y^{\text{len}(\varphi(n))}(\varphi(n)) $$ is recursive.
Using TMs this is quite straightforward: just simulate the TM that computes $\varphi$ and print only the $y$-th term of that TM executed with input $n$. I am interested in a purely "recursion-theoretical" proof, i.e., I would like to know how that function can be written in terms of primitive recursion and/or $\mu$-recursion, which I'm finding tricky.
