I want to show that these two forms of $AC$ are equivalent:
$(1)$ For each collection of nonempty sets $X$ there is a choice function.
$(2)$ For each collection of pairwise disjoint nonempty sets $X$ there is a choice function
$(1) \Rightarrow (2)$ is trivial.
$(2) \Rightarrow (1)$: Let $X=\{X_i: i\in I\}$ be a collection of nonempty sets with index set $I$. Then $X':=\{X_i \times \{i\}:i\in I\}$ is a collection of pairwise disjoint sets. But how can I get a choice function from here? I've read that a set which has exactly one element in common with each element of the collection would be a choice function, but I kinda don't understand why this set can be seen as a function.
Once you have a choice function from $\{X_i\times\{i\}\mid i\in I\}$, meaning some $f$ whose domain is $I$ and $f(i)=\langle x,i\rangle$ where $x\in X_i$, simply take the projection onto the left coordinate, and show that this is a choice function for the original family.