What does the set-builder notation with ∀n mean for membership?

[cappadonza]

Hi this is more of a set theory question really, I'm a bit confused,

say $$\mathcal{F}$$ is collections of sets, and $$\mathcal{F}_n$$ is a sequence of sub collections of sets and say $$B_{1}, B_{2} ...$$ is a sequence of sets
what does the following mean $$\mathcal{S} = \{ A \in \mathcal{F} \colon A \cap B_{n} \in \mathcal{F}_n \forall n \in \mathbb{N} \}$$

for an element to be a member of the set $$\mathcal{S}$$ which of the conditons must be statisfy
does this mean if $$A \cap B_{1} \in \mathcal{F}_1 , \space A \cap B_{2} \in \mathcal{F}_2 ...$$ then it belongs to the set or
does it mean all these conditons must be met of it to be a members of the set $$A \cap B_{1} \in \mathcal{F}_1 , \space A \cap B_{1} \in \mathcal{F}_2 , \space A \cap B_{1} \in \mathcal{F}_3 ...$$ for each $$B_{1}, B_{2} ...$$

 

[EnumaElish]

The first is the correct interpretation, if $A \cap B_1\in\mathcal{F}_1$ and $A \cap B_2\in\mathcal{F}_2$, etc., for all n = 1, 2,..., then A is in the collection.