Let $[n]$ denote $\{1,2,\dots n\}$.
Show there is a bijection between $[m]\times[n]$ and $[mn]$.
I was thinking about doing something related to induction to prove that there is some value $b$ for every $f(a)$. Then probably settle for the definition of a product to say that in $\{(a,b)\mid a\in A ; b\in B\}$ so there can only be values corresponding from $A$ and $B$ in $(a,b)$. If this seems correct, let me know.
If there's anyone with a more rigorous proof, I would be delighted.