I take it that you're using the letter phi to stand for the empty set.

As far as I can tell, you must have left out something in the statement of the problem.

The following is NOT a theorem (of, say, set theory), so I don't know why you would be asked to prove it [edit: '0' for the empty set]:

If f:X->Y, then f(0) = 0.

Counterexample:

Let X = {0}, let Y = {1}, let f = {<0 1>}.

Then, also, your "proof" makes no sense. There's no basis to assume there is an x1 such that x1 is in 0 and/or f(x1) in 0.

What book is this problem taken from?