TITLE: Is the powerset of every Dedekind-finite set Dedekind-finite?
QUESTION [5 upvotes]: Is the powerset of every Dedekind-finite set Dedekind-finite? 
I think this statement can be written in $\textbf{Set}$: If every mono (=injection) $f: A \to A$ is iso (=bijection), then every mono $g: 2^A \to 2^A$ is iso.
(Please edit if necessary)
Does the answer depend on some Choice principle? 
Off-hand I don't see a referent to this. The Wikipedia entry for Dedekind-infinite only states that the powerset of a D-infinite set is D-infinite. (I tried searching Math.SE with the obvious keywords and faced ~5k entries).

REPLY [5 votes]: The following is implicit in the answers given by Andres and Asaf, but it looks simpler, so it might be worthwhile to make it explicit.  Let $X$ be any infinite set, possibly Dedekind-finite.  Let $F$ be the function with domain $\omega$ defined by assigning to each natural number $n$ the family of all $n$-element subsets of $X$.  Then $F$ is a one-to-one function from $\omega$ into $\mathcal P(\mathcal P(X))$, the power set of the power set of $X$.  So if $X$ is D-finite, then either $X$ itself or $\mathcal P(X)$ is a D-finite set with D-infinite power set.