Here is the problem.
Suppose and are nonempty sets and is a function. Let and be subsets of . Prove that
We do this by showing that and
So, for the first part:
Suppose . Then for some , , and .
or [Note: throughout this proof 'or' means 'inclusive or'. So or means or or both.
For the second part:
Then for some , and or
So, from (1) and (2):