Image of Intersection under Injection/General Result

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Theorem

Let $S$ and $T$ be sets.

Let $f: S \to T$ be a mapping.

Let $\mathcal P \left({S}\right)$ be the power set of $S$.


Then:

$\displaystyle \forall \mathbb S \subseteq \mathcal P \left({S}\right): f \left[{\bigcap \mathbb S}\right] = \bigcap_{X \mathop \in \mathbb S} f \left[{X}\right]$

if and only if $f$ is an injection.


Proof

An injection is a type of one-to-one relation, and therefore also a one-to-many relation.


Therefore Image of Intersection under One-to-Many Relation applies:

$\displaystyle \forall \mathbb S \subseteq \mathcal P \left({S}\right): \mathcal R \left[{\bigcap \mathbb S}\right] = \bigcap_{X \mathop \in \mathbb S} \mathcal R \left[{\mathbb S}\right]$

if and only if $\mathcal R$ is a one-to-many relation.


We have that $f$ is a mapping and therefore a many-to-one relation.

So $f$ is a one-to-many relation if and only if $f$ is also an injection.


It follows that:

$\displaystyle \forall \mathbb S \subseteq \mathcal P \left({S}\right): f \left[{\bigcap \mathbb S}\right] = \bigcap_{X \mathop \in \mathbb S} f \left[{X}\right]$

if and only if $f$ is an injection.

$\blacksquare$