Properties of injective functions

Hi.

**problem:**

Show that if is injective and , then .

I worked with this for a while but was unable to come up with a good answer.

After a bit of searching, I fould an article on planetmath which answers this question, but I do not understand it completely.

Here's what it says:

Theorem. Suppose is an injection. Then, for all , it is the case that .

Proof. It follows from the definition of that , whether or not happens to be injective.

Hence, all that need to be shown is that .

Assume the contrary. Then there would exist such that .

By definition means , so there exists such that .

Since is injective, one would have , which is impossible **because is supposed to belong to ** but is not supposed to belong to .

I do not see why is supposed to belong to . What am I missing here?

Thanks!