What is the relation between the following two principles?
$$ AC(X): \quad \text{ for every } \mathcal{F} \subset \mathcal{P}(X)\backslash\{\emptyset\} \text { there is a choice function} $$
$$ AC_{|X|}(X): \quad \text{ for every } \mathcal{F} \subset \mathcal{P}(X)\backslash\{\emptyset\} \text { s.t. } |\mathcal{F}|=|X| \text{ there is a choice function} $$
Obviously the former implies the latter. The intuition that makes plausible an equivalence between the two principles is that, since every set $F\in\mathcal{F}$ is a subset of $X$, the number of different choices cannot be too large...in particular we cannot make more than $|X|$ different choices (hence if the family is larger then we must necessarily choose the same element from different sets). Needless to say, I tried to prove it for a while but without much luck. Of course I am not assuming $X$ to be well-orderable (otherwise that would be easy). In my specific context I have $X=\mathbb{N}^\mathbb{N}$, but I am curious to know what is the answer for a general $X$.