
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).

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

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!