Page "Axiom of choice" Paragraph 6

Each choice function on a collection X of nonempty sets is an element of the Cartesian product of the sets in X.

This is not the most general situation of a Cartesian product of a family of sets, where a same set can occur more than once as a factor ; however, one can focus on elements of such a product that select the same element every time a given set appears as factor, and such elements correspond to an element of the Cartesian product of all distinct sets in the family.

The axiom of choice asserts the existence of such elements ; it is therefore equivalent to: