Math Help - Function Proof

1. Function Proof

Suppose f: A--> B and let C be a subset of A.

Prove: if f is injective, then f^(-1)[f(C)] = C

I know from a previous theorem that C is a subset of f^(-1)[f(C)] so need only to prove f^(-1)[f(C)] is a subset of C.

So far I have:

Let f be injective. Let x be an element of f^(-1)[f(C)] . Then f(x) is an element of f(C).

2. Then by definition of $f(C)$ we have $\left( {\exists z \in C} \right)\left[ {f(z) = f(x)} \right]$.
Now what does injectivity tell us?

3. Since f is injective x = y and it then follows that x as an element of f^(-1)[f(C)] makes f^(-1)[f(C)] a subset of C.

Thanks, Plato!