Let be a mapping. We will prove that , with equality when is injective. Note that does not have to be closed, open, or even continuous for this to be true. It can be any mapping.
Let . The mapping of in is . As for , it may overlap with , we the mapping be not be injective. Hence, .
>Taking on both sides, we get .
How can we take the inverse on both sides and determine this fact? Is the reasoning valid? Yes. All the points in that map to also map to . However, there may be some points in that do not map to .
Are there other analogous points about mappings in general? In , select two sets and such that . Then