Question on proofs on functions (discrete)

Question

Having trouble figuring out this proof:

Let A --> B.

Let C, D c E.

For any set C, let f(C) = { f(c) | c∈C }.

Prove that f(C U D) = f(C) U f(D)

Answer 1

To show that two sets S,T are equal you need to show that S is a subset of T and T is a subset of S. In our case S=f(C union D) and T=f(C) union f(D). For example let y be an element of the set f(C union D). This means that there exists an x in the set (C union D) such that y=f(x). But since x belongs to (C union D) we must have that either x is an element of C or x is an element of D. But in the first case we have that y is an element of f(C) and in the second case we have that y is an element of f(D). Hence, we have shown the relation f(C union D) is a subset of f(C) union f(D). Similarly we can show the other direction and obtain that the sets f(C union D) and f(C) union f(D) are equal.