Probability: Intersection and Union of sets I want to show the following statement,
Let $(\Omega, \mathcal{F}, \mathbb{P})$ be a probability space. Let $E_1,\dots , E_n \in \mathcal{F}$. Let, $$\mathcal{C}=\left\{F \in \mathcal{F}: F = \cap_{i=1}^{n}F_i, F_i \in \left\{E_i, E_i^c\right\}, i = 1, . . . , n\right\}$$
For all $1\leq i_1 < \dots < i_k\leq n$ , where $k = 2, \dots , n − 1$, show that $\cap_{i=1}^{k}E_{i_j} = \cup\left\{F ∈ \mathcal{C} : F \subset \cap_{j=1}^{k}E_{i_j}\right\}$.
I don't see how to visualize the following set $\cup\left\{F ∈ \mathcal{C} : F \subset \cap_{j=1}^{k}E_{i_j}\right\}$.
Any ideas or leads?
Thanks :)

I will give an answer based on @user940347 idea.
Let $I=\left\{1,\dots,n\right\}$, $I_k=\left\{ i_1,\dots,i_k\right\}$ and $\tilde{I}_k=I\setminus I_k$. We have the following,
\begin{align*}
    \cap_{j=1}^kE_{i_j}&=\cap_{i\in I_k}E_i\\
    &= \bigcup\left(\cap_{i\in I_k}E_i\cap_{j\in\tilde{I}_k} 
    \tilde{F}_j\right) \quad \text{where $\tilde{F}_j\in\left\{E_j,E_j^c\right\}$}
\end{align*}
We have $\left(\cap_{i\in I_k}E_i\cap_{j\in\tilde{I}_k}\tilde{F}_j\right)\in\mathcal{C}$ by definition of $\mathcal{C}$ and of course $\left(\cap_{i\in I_k}E_i\cap_{j\in\tilde{I}_k}\tilde{F}_j\right)\in \cap_{i\in I_k}E_i$ by definition of intersection.
Therefore the $\subset$ is proved.
