If we have three sets $A,B,C \subset\Omega$ and we take the intersection with another set $Y$, we have $3$ sets i.e. $A\cap Y,B\cap Y$ and $C\cap Y$
My question is how can we prove that the intersection with a set doesn't affect the indexation of the initial set (here we have $3$ initial sets) ?
More formally if we take $E_\alpha\in Image(\mathscr P(\Omega) ^{E})$ where $E$ is finite, infinite, or infinite uncountable, how can we know that $\{E_\alpha \cap Y\}_{\alpha \in E}$ has the same cardinality (I'm not sure if it's the correct word) as $\{E_\alpha \}_{\alpha \in E}$ ?