Show that the cardinality of a power set of natural numbers is exactly equal to the map of a set of natural numbers to another, which is $|\mathcal{P}(\mathbb{N})| = |\operatorname{Maps}(\mathbb{N},\mathbb{N})|$.
I cannot find anything to start with other than knowing there is an injective map from $\mathbb{N}$ to $\mathbb{N}$
By Cantor-Bernstein, it's enough to find (1) an injection from $\mathcal{P}(\mathbb{N})$ into $Maps(\mathbb{N}, \mathbb{N})$, and (2) an injection from $Maps(\mathbb{N}, \mathbb{N})$ into $\mathcal{P}(\mathbb{N})$.
Towards (1): Suppose I have a set $X\subseteq\mathbb{N}$. This can be viewed as a map from $\mathbb{N}$ to $\{Yes, No\}$ - you ask "is $n\in X$?" and I output "Yes" or "No" accordingly. Does this help?
Towards (2): This one's a bit harder, but: a function $F$ is completely defined by its graph, the set $\{(a, b): F(a)=b\}$. So . . .