Trouble with unique existance proof

Exercise 3.7.1 of "How To Prove It" (Vellemen 2006)

Suppose F is a family of sets. Prove that there is a unique set A that has the following two properties:

(a)

(b)

So what I want to show is:

I have the existence part down:

Suppose . Then .

Suppose and and such that .

, so clearly . Since then it

follows that , therefore . Since , and

were arbitrary, .

Where I'm having trouble is with the uniqueness proof. I believe that in order to prove uniqueness,

I have to show that

The first half of proving the equality is trivial, that is so clearly . It's the

other half that is giving me trouble, that is how do I show that ? I can see that it is true but for some reason I can't come up with well reasoned argument as to why.