Rooks on chessboard

I place some rooks on a chessboard (possibly more than one rook on some squares). Given:

* There is at least one rook on each rank.
* There is at least one file without a rook on it.

Prove: there are a file f and a rank r such that there is at least one rook at the intersection of f and r, and the number of rooks on f is bigger than the number of rooks on r.

Hm... i might have seen this somewhere. It sounds like the kind of problem that could be solved by counting something in two ways. In this case that would be the rooks. Working on it...