Working on a proof and I need some help

Let f : A ----> B be an injective (1:1) function and consider C a subset of A. Show that f^(-1)(f(C)) = C or [inverse image of f(C) = C]

e = an element of

What I have so far is...

C = { a e A such that a e C}

f(A) = { b e B such that there exists a e A, for f(a) = b}

f(C) = { b e B such that there exists c e A, for f(c) = b}

f(C) is a subset of B

f^(-1)(f(C)) = { a e A such that f(a) e f(C)}

Am I doing this right so far? And if I am, how do I go from the "f(a) e f(C)" part to "a e C" which would then be equal to C and my proof will be done?

I also have a similar problem to this except f : A ----> B is surjective (onto) rather than injective. How does this change the proof? Very lost...

- Nicole