Recursive Function is Arithmetically Definable

Theorem
Let $f: \N^k \to \N$ be a recursive function.

Then there exists a $\Sigma_1$ WFF of $k + 1$ free variables:
 * $\map \phi {y, x_1, x_2, \dotsc, x_k}$

such that:
 * $y = \map f {x_1, x_2, \dotsc, x_k} \iff \N \models \map \phi {\sqbrk y, \sqbrk {x_1}, \sqbrk {x_2}, \dotsc, \sqbrk {x_k} }$

where $\sqbrk a$ denotes the unary representation of $a \in \N$.

Proof
By definition of recursive function:
 * $f$ can be obtained from basic primitive recursive functions using the operations of:
 * substitution
 * primitive recursion, and
 * minimization on a function
 * a finite number of times.

The existence of $\phi$ follows from:
 * Basic Primitive Recursive Functions are Arithmetically Definable
 * Substitution of Arithmetically Definable Functions is Arithmetically Definable
 * Primitive Recursion on Arithmetically Definable Function is Arithmetically Definable
 * Minimization of Arithmetically Definable Function is Arithmetically Definable