Let $n$ be a non-zero natural number. We set
$$\Phi_n : \left|\, \begin{aligned} \{0,1\}^n &\rightarrow \llbracket 0, 2^n - 1 \rrbracket \\ (x_j)_{j \in \llbracket 1,n \rrbracket} &\mapsto \sum_{j=1}^{n} x_j 2^{n-j} \end{aligned} \right.$$
and $A_n = \left\{ \sum_{j=1}^{n} x_j 2^{n-j},\, (x_j)_{j \in \llbracket 1,n \rrbracket} \in \{0,1\}^n \right\}$.
Specify $\operatorname{Im} \Phi_n$ as a function of $A_n$.