Inverse Image Mapping of Relation is Mapping

Theorem
Let $\mathcal R \subseteq S \times T$ be a relation on $S \times T$.

Let $\mathcal R^\gets$ be the mapping induced on the power set of $S$ by the inverse relation $\mathcal R^{-1}$:


 * $\mathcal R^\gets: \mathcal P \left({T}\right) \to \mathcal P \left({S}\right): \mathcal R^\gets \left({Y}\right) = \mathcal R^{-1} \left[{Y}\right]$

Then $\mathcal R^\gets$ is indeed a mapping.

Proof
$\mathcal R^{-1}$, being a relation, obeys the same laws as $\mathcal R$, and so Mapping Induced on Power Set is Mapping applies directly.

Comment
Note that it is not necessarily the case that $\left({\mathcal R^\to}\right)^{-1} = \mathcal R^\gets$.

Look closely at the notation:
 * The first is the inverse of the mapping induced by the relation.
 * The second is the mapping induced by the inverse of the relation.