The original question is (from Mathematical Logic, A course with Exercises):
Assume Axiom of Choice, let $a$ and $b$ be 2 infinite sets with $\text{card}(a)=\lambda$ and $\text{card}(b)=\mu$. Assume $\lambda > \mu$. Let $g$ be an injective mapping from $b$ to $a$.
Determine the cardinality of $\{f\in b^a: \text{card}(f^{-1}(b))= \lambda\}$
I know this is the cardinality of $\mu^\lambda$, but the solution manual says $\mu^\lambda = 2^\lambda$. I think I have missed some lemma. How to prove it?
Thanks for any help!