Let consider the Vitali set $V \subset \mathbb R$, which is constructed using the axiom of choice. (I could take any other mathematical "object" that can be constructed using the axiom of choice, but I chose the Vitali set just to make the purpose clearer).
On the one hand, the Vitali set exists thanks to the axiom of choice. On the other hand, the power set $\mathcal P (\mathbb R)$ exists in ZF without assuming AC. In some sense, the existence of any $A \in \mathcal P (\mathbb R)$ is independant on (i.e. does not require) AC.
But then, I had the following reflection : the existence of the Vitali set $V \in \mathcal P (\mathbb R)$, as an element of this power set, should not require the axiom of choice... There must be some fallacy here ; that's why I would like some clarifications.
To be concise : can we say that the axiom of choice "affects" the existence of elements in $\mathcal P (\mathbb R)$ (as it seems to be the case for the existence of the Vitali set), although the construction of $\mathcal P (\mathbb R)$ (in ZF) has nothing to do with AC?