Did you just not understand this part or the whole proof in general? Let me know if the latter is the case :).
So, what they mean is: For all sets X and Y, being subsets of A (which is again a subset of the powerset of F which includes all possible subsets of F) we can prove that:
X is a subset of Y if and only if f(X) is a subset of f(Y).
Why is that true?
Well, Let us prove each direction:
X subset Y implies f(X) subset f(Y)
So, if X is a subset of Y, every element of X is part of Y (but not necessarily the other way around).
Now, what is f(X)? Well, if X={x,y,…} (subset of P(F)) then f(X) = {f(x),f(y),…} (subset of is P(n) which is the power set of all natural numbers up to n). Now, if there is any element z in Y but not in X, then f(Y)= {f(x),f(y),…, f(z)} superset of {f(x),f(y),…} =f(X)
First part done :)
Other direction:
f(X) subset of f(Y) implies X subset of Y
If any element f(a) is both in f(X) AND f(Y), then obviously a is both in X and Y. Let k (natural number) be in f(Y) but not f(X). As f is a one to one mapping of F to the natural numbers 1,….,n there is an element z in F such that f(z)=k.
Then, z has to be in Y, but not in X (as f is one to one).
Hence it follow that X is a subset of Y