Intersection of Primitive Recursive Sets

Theorem
Let $A, B \subseteq \N$ be subsets of the set of natural numbers $\N$.

Let $A$ and $B$ both be primitive recursive.

Then $A \cap B$, the intersection of $A$ and $B$, is primitive recursive.

Proof
$A$ and $B$ are primitive recursive, therefore so are their characteristic functions $\chi_A$ and $\chi_B$.

Let $n \in \N$ be a natural number.

Then $n \in A \cap B \iff \chi_A \left({n}\right) = 1 \land \chi_B \left({n}\right) = 1$.

So:

Thus $A \cap B$ is defined by substitution from the primitive recursive functions $\operatorname{mult}$, $\chi_A$ and $\chi_B$.

Hence the result.