This is still the pigeonhole principle, just inverted - If you have more holes than pigeons (and you can’t make fractional or quantum pigeons) at least one hole must have no pigeons.
It's more than similar, it's gone from talking about the necessary properties of a function with an image smaller than its domain, to the impossibility of an inverse function as the image cannot be larger than its domain.
We might as well call this "the whole-pigeon principle."