We use the fact that the choice axiom is equivalent to the existence of a choice function (see page 59).

Given the definition of the cartesian product on page 113, the product is not empty if there is at least one function $\mathbf{x}:J\rightarrow\cup_{\alpha\in J}A_{\alpha}$ such that for every $\alpha\in J$ , $x_{\alpha}\in A_{\alpha}$ , but this just says that there is a choice function (see page 59). More formally, we would have to consider a bijective function $h:J\rightarrow\{A_{\alpha}\}$ (see page 36), a set $\mathcal{B}=\{A_{\alpha}\}=\{B|B=h(\alpha)\mbox{ for some }\alpha\in J\}$ and $\mathbf{x}':\mathcal{B}\rightarrow\cup_{B\in\mathcal{B}}B$ , and then define $\mathbf{x}=\mathbf{x}'\circ h$ , where the existence of $\mathbf{x}'$ is ensured by the choice axiom.

Vice versa, for every collection $\mathcal{B}$ of nonempty sets, let $J=\mathcal{B}$ and define the indexing function $h:\mathcal{B}\rightarrow\mathcal{B}$ by $h(B)=B$ . Then, the product $\prod_{B\in\mathcal{B}}B$ is not empty, and there is at least one function $\mathbf{x}:\mathcal{B}\rightarrow\cup_{B\in\mathcal{B}}B$ such that for every $B\in\mathcal{B}$ , $\mathbf{x}(B)\in B$ , but then $\mathbf{x}$ is a choice function.