$A\subset N^k$ is recursively enumerable if and only if $A$ is the domain of a recursive relationship
I am trying to prove this equivalence for the characterization of an r.e set. I am working on the book "The Incompleteness Phenomenon" by Goldstern and Judah.
This is my definition for a r.e set, (the definition that I should use to solve the problem):
$B\subset N^k$ is said to be r.e if and only if there is an $\Sigma^0_1$-formula $\phi(x_1,...,x_k)$ such that $B=\{(n_1,...,n_k)\in N^k : N\Rightarrow\phi(n_1,...,n_k) \}$.
I try to attack the problem using the definitions, but I don't have clear ideas. When I try to understand how to handle r.e sets I always find developments using the theory of recursion from the perspective of computing (I found this proposition in my mathematical logic class); I would appreciate any sketch or reference to check.
Thank you in advance.