Prove that the intersection of definable sets is definable

Question

Hello I have a question :

$F$ is a family of definable sets. Prove that the intersection of all the sets in the family is definable. ($F$ could be infinite)

Definition (Definable): a set $K$ of assignments is definable if there is a set of formulas A such that $\operatorname{Ass}(A) = K$.

I don't know how to approach it because F could be infinite.. I will be glad to get help.

Answer 1

$\forall A\in F$ there is a set of formulas $\Sigma_A \space s.t \space Ass(\Sigma_A)=A$ (because A is definable).
Now, lets define $\Sigma= \cup_{A\in F} \Sigma_A$.

Prove that : $Ass(\Sigma)=\cap_{A\in F} A$