How we prove in a function f that f(R)=R?
By proving that is '1-1'?
If , i.e., it is declared that the domain and the codomain of are , then the claim that means that is surjective. On the other hand, the fact that is 1-1 means that it is injective. These are different concepts; none of them implies the other. For example, for we have , but is not 1-1.