Verify the following identity with families of sets

($\cap \mathcal{F}$ ) $\cap $ ($\cap \mathcal{G}$) = $\cap $ ( $ \mathcal{F} \cup \mathcal{G} $ )

Question

Verify the following identity with families of sets

($\cap  \mathcal{F}$ ) $\cap $ ($\cap \mathcal{G}$) = $\cap $ ( $ \mathcal{F} \cup \mathcal{G} $ )

empty · Answer

what is any of this

anonymous · Answer

F and G are supposed to be families. Just they arent indexed.

anonymous · Answer

Families of sets

anonymous · Answer

I guess $\cap \mathcal{F} $ be equivalent to $\cap_{ i \in I} A_{i} $, but verifying the identities with these are different since I could make claims in regards to the index I.

anonymous · Answer

Here's how I started, though. 
$ x \in $ ($\cap \mathcal{F}$ ) $\cap $ ($\cap \mathcal{G}$)
iff ( $x \in \cap \mathcal{F}$  )  $\cap $ ( $x \in \cap \mathcal{G}$ )
iff $\forall A \in \mathcal{F} $ ( $x \in A $ ) $\cap $ $\forall A \in \mathcal{G} $ ( $ x \in A$ )

I don't know if I can go any further on that end of it. I didnt really like how the other side of the equality was coming apart, I kept getting stuck