The dimension of vector space $F^{X}$.

Question

Here's the problem:

Let $F$ be field, $X$ an infinite set and $F^{X}$ be the set of all functions $f:X\rightarrow F$. Then $F^{X}$ is a vector space over $F$ (with $(f+g)(x)=f(x)+g(x)$ and $(rf)(x)=rf(x)$ ).

The thing puzzling me is how to prove the equation $\dim_{F}F^{X}=|F|^{|X|}$ because I find it difficult to determine some basis of $F^{X}$ directly. Anyway, any and all help is appreciated. Thanks.