elementary set theory - Is the sets of all maps from $mathbb{N}$ to $mathbb{N}$ countable?

Is the sets of all maps from $\mathbb{N}$ to $\mathbb{N}$ countable?

Attempts: I think it is uncountable. Consider $P(\mathbb{N})$ where $P(A)$ denote as the power sets of A. We know that $P(\mathbb{N})$ is uncountable. Moreover, by the axiom of choice, for any non empty set, we know that there exist a function $f: P(\mathbb{N})\to \mathbb{N}$ . Note that $P(\mathbb{N})$ have uncountable many elements and since each $f$ belongs to the set of all mapping from $\mathbb{N}$ to $\mathbb{N}$ so there exist infinite uncountably elements and hence this set are uncountable. Is my attempt work??