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 would an article on planetmath which answers this question, but I do not understand it completely.

Here's what it says:

Suppose Theorem.

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!