Preimage of Intersection under Mapping/General Result

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Theorem

Let $S$ and $T$ be sets.

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

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

Let $\mathbb T \subseteq \mathcal P \left({T}\right)$.


Then:

$\displaystyle f^{-1} \left[{\bigcap \mathbb T}\right] = \bigcap_{X \mathop \in \mathbb T} f^{-1} \left[{X}\right]$


Proof

$f$ is a mapping.

Therefore it is by definition a many-to-one relation.

It follows from Inverse of Many-to-One Relation is One-to-Many that its inverse $f^{-1}$ is a one-to-many relation.


Thus Image of Intersection under One-to-Many Relation: General Result applies:

$\displaystyle \mathcal R \left[{\bigcap \mathbb T}\right] = \bigcap_{X \mathop \in \mathbb T} \mathcal R \left[{X}\right]$

where here $\mathcal R = f^{-1}$.

$\blacksquare$


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