Properties of injective functions
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?