|{f:A--> A}| = |{f:A--> 2}| for A infinite. Why?

How does one prove that, for an infinite set A, that $\displaystyle |A|^{|A|} $=$\displaystyle 2^{|A|} $, that is, that the set of all functions from A to A has the same cardinality as the set of all functions from A to {0,1}, or equivalently the power set of A?

The moderator asked posters to indicate the proof system in which one is working. The usual set-theoretical methods are fine. I am not working in the constructive universe. If singular cardinals give problems, assume that |A| is a regular cardinal.