Proposition 4

Let $(X_\iota)_{\iota \in I}$ be a family of sets, and let u be a bijection of a set K onto the index set I. If U is the graph of u, the mapping $F \to F \circ U$ of $\prod_{\iota \in I} X_\iota$ into $\prod_{x \in K} X_{u(x)}$ is bijective.

Let

(§4, Proposition 1). The mapping $F\to F\circ U$ ($F\in A^I$) is a bijection of $A^I$ onto $A^K$ (Proposition 2). It is evident that the condition “for each $\iota\in I$, $F(\iota)\in X_\iota$” is equivalent to “for each $x\in K$, $(F\circ U)(x)\in X_{u(x)}$”, and the result follows.