How Is This a Bijection from $P(X)$ to $2^X$?

Question

Let $X$ be a set, $P(X)$ the power set of $X$, and $2^X$ the set of all function from $X$ to $\{0, 1\}$.
Now define $J: P(X) \rightarrow 2^X$ by $J(A) = \chi_A$, where $\chi_A$ is the characteristic function.
Show that $J$ is a bijection from $P(X)$ to $2^X$.

I can demonstrate that $J$ is a bijection from $P(X)$ to $\{\chi_A | A \subset X\}$.
But how do I know for sure that $\{\chi_A | A \subset X\} = 2^X$?
Or, is there a function in $2^X$ that does not take the form of $\chi_A$?
I am not seeing how the answer is no, but I can't think of any counter examples either.

Answer 1

Let $\varphi \in 2^X$ and set $A = \varphi^{-1}(1)$, then $\varphi = \chi_A$. Hence $2^X = \{\chi_A \mid A \subseteq X\}$.