$\{(X,Y) | X \cap Y = Z \} = \{X | Z \subset X\} \times \{Y | Z \subset Y\}$?

Question

I want to prove question Sum of cardinals of all intersections of finite set $E$.

is $\{(X,Y) | X \cap Y = Z \} = \{X | Z \subset X\} \times \{Y | Z \subset Y\}$ where $Z \subset E$ and $X, Y \in \mathscr P(E)$?

Answer 1

You actually have that $$ \{(X,Y) : Z = X\cap Y\}\subset \{(X,Y) : Z\subset X\cap Y\} = \{X : Z\subset X\}\times\{Y : Z\subset Y\}. $$

Answer 2

No. The set on the right is much bigger in general.

Answer 3

No. For example, $\{1\} \subseteq \{1,2\}$ and $\{1\} \subseteq \{1,2,3\}$, but $\{1,2\}\cap\{1,2,3\} = \{1,2\} \ne \{1\}$.