How could you go about proving the following:
Let $f_1$ and $f_2$ denote bijections where $f_1:A_1 \rightarrow B_2$ and $f_2:A_2 \rightarrow B_2$. If $g:A_1 \times A_2 \rightarrow B_1 \times B_2$ where $g(x, y) = (f_1(x), f_2(y))$, then $g$ is a bijection.
This is a sample exam problem. I'm not really sure how to go about proving this, as it seems intuitively correct to me, so I'm not really sure where to start.
I'm not so much interested in a proof as some sort of starting point or hint that could tell me what I need to prove or what I might be able to use.
EDIT: Just to be clear, we're using what I presume is the standard definition for a bijection, a function that is both an injection (one-to-one) and a surjection (onto).