Proving Dedekind Finiteness of Set of Functions. I'm struggling with a question from a past paper for a set theory exam. Can't really see a way forward as the different types of finite make it tricky.
The question is 'Let A be finite and B be Dedekind finite. (Where 'finite' means Kuratowski finite, ie every Kuratowski inductive set for A contains A. And Dedekind finite is that every injection from B to B is a bijection). Show that the set of functions from A to B is Dedekind Finite.'
I was thinking the best strategy is to let F be an injection from the set of functions to itself and then assume there is an element that it doesn't hit and find a contradiction. But the different types of finite-ness is making it hard for me to see where to start.
Any advice?

Really what you're being asked to prove is that the finite product of Dedekind finite sets is Dedekind finite again.
With induction it is enough to show that the product of two Dedekind finite sets is Dedekind finite.
So now suppose that $X$ and $Y$ are Dedekind finite. If $X\times Y$ is not Dedekind finite, then it has a countably infinite subset (recall that a set is Dedekind finite if and only if it does not have a countably infinite subset).
Let $N$ be such countably infinite subset of $X\times Y$. Ask yourself what could $M=\{x\in X\mid\exists y(\langle x,y\rangle\in N)\}$ and $N_x=\{y\in Y\mid\langle x,y\rangle\in N\}$ (for $x\in X$) be. They can be finite, or countably infinite. Now derive a contradiction from either option.
