proof that a relation is union of functions if and only if AC
Theorem.
A relation is the union of a set of functions
each of whichhas the same domain as if and only if for each set ofnonempty sets, there is a choice function on .
Proof.
Suppose that is a relation with . Let be givenby . There is be a choice function on . Let , and for each pair, let send to and agreewith elsewhere. Let . Clearly , so suppose; then there is a pair such that . Either , or . In each case, .Thus, For each pair , , so . Therefore, .
Suppose that is set of nonempty sets. Let . A set is an element of if and only if for some . Thus,. There is a set of functions, each of which hasdomain , such that . Let ; then, and for each pair ,; i.e., .Each such is, thus, a choice function on .∎