Question

What is the meaning of "It’s obvious that ∀X, Y ∈ A(X ⊂ Y ⇐ ⇒ f(X) ⊂ f(Y )); A has a ⊂-maximal element"?

Answer 1

So, as A is a subset of P(F), it is made of some subsets of F of (potentially varying) but FINITE length, as F is finite.

Now, as B has a setsize-maximal element we will let that element be f(Z) for some set Z element of A.

That together with the statement in brackets implies that any other set X in A for which f(X) is a subset of f(Z) is hence a subset of Z.

That means Z is a setsize-maximal element of A.