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.