proving that f(x) > g(y)

We are given sets A, B, C, and D, with x in A and y in C, and asked to prove that f(x) > g(y).
It is sufficient if to be able to do the following:
1. prove that x is in A implies f(x) is in B;
2. prove that if y is in C, then g(y) is in D;
3. prove that B > D (that is, every element of B is greater than every element of D).
keywords: Mathematics, Logic, proof technique