Proposition 7

Let $(\mathrm{X}_\iota)_{\iota \in \mathrm{I}}$ be a family of sets whose index set I is not the empty set. Let $(\mathrm{J}_\lambda)_{\lambda \in \mathrm{L}}$ be a partition of I. Then the mapping

of $\prod\limits_{\iota \in \mathrm{I}} \mathrm{X}_\iota$ into the product set $\prod\limits_{\lambda \in \mathrm{L}} \Big(\prod\limits_{\iota \in \mathrm{J}_\lambda} \mathrm{X}_\iota\Big)$ is bijective ("associativity" of products of sets).

From the interpretation of $\mathrm{pr}_{\mathrm{J}_\lambda} f$ as the graph of the restriction of the function whose graph is $f$ (no. 4), it follows that the statement that the mapping $f \to (\mathrm{pr}_{\mathrm{J}_\lambda} f)_{\lambda \in \mathrm{L}}$ is bijective means that, for each family $(v_\lambda)_{\lambda \in \mathrm{L}}$, where $v_\lambda$ is a mapping of $\mathrm{J}_\lambda$ into $\bigcup\limits_{\iota \in \mathrm{I}} \mathrm{X}_\iota$, there exists a unique mapping $u$ of I into $\bigcup_{\iota \in I} X_\iota$ such that $v_\lambda$ is the restriction of $u$ to $J_\lambda$ for each $\lambda \in L$. But this is a consequence of the hypothesis that $(J_\lambda)_{\lambda \in L}$ is a partition of I (§4, Proposition 8).

¶ The bijection defined in Proposition 7, and its inverse bijection are said to be canonical.

Remarks