elementary set theory - Bijection between $[mathbb Nto {0,1}]$ and $[mathcal P(mathbb N) to {0,1}]$

In ZFC (edit : and other axiomatic systems), does there exists a bijection between $[\mathbb N\to \{0,1\}]$ and $[\mathcal P(\mathbb N) \to \{0,1\}]$?

Extrapolating from https://en.wikipedia.org/wiki/Algebraic_normal_form it seems that there is exactly $2^{2^n}$ functions from a set where elements are described with n bits to $\{0,1\}$

So by imagining what the limit would be for all possible integer size, it seems that an infinite string of $\{0,1\}$ is exactly the description of one particular function from N to $\{0,1\}$

But then with the same argument, an infinite string of $\{0,1\}$ would also be the description of exactly one particular function from a subset of $\mathbb N$ (which also happens to be a function from $\mathbb N$ to $\{0,1\}$ in that view expressed here) to $\{0,1\}$

Hence i would tend to conclude that $[\mathbb N\to \{0,1\}]$ and $[\mathcal P(\mathbb N) \to \{0,1\}]$ are the same thing ..

What is the view of ZFC (and other axiomatic systems) on the matter, do both set of functions have the same cardinality (which i take is equivalent to ask for the existence of a bijection between the two ) ?

Edit : i would like to extend the questions to other axiomatic systems than ZFC