elementary set theory - Let $alpha, beta, gamma$ be cardinals, $beta leq gamma$, prove $alpha ^{beta}le alpha ^{gamma}$

Let $|A|=\alpha, |B|=\beta, |C|= \gamma$ be cardinals and $\beta \leq \gamma$. Prove $\alpha ^{\beta}\le \alpha ^{\gamma}$.

So from the given we know that there's an injection $f:B\to C$ and some functions $h:B\to A, g: C\to A$. We want to prove there's an injection $l_1:A\to C$. It appears that $f$ doesn't help here.

Trying to take representatives from $A$ and show they're in $C$ and there's an injection doesn't work so maybe the function should be $l_2: h \to g$ but I don't know how to work with it.

Answer

Given the injection $f$, for each function $h: B \to A$ you can associate a $g(y): C \to A$ by $g(f(x))=h(x)$ if $y \in f(B),$ otherwise $g(y)=$something in $A$ Since $f$ is an injection, the $g$'s will be distinct whenever the $h$'s are.