Inverse Image Mapping of Relation is Mapping

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

Then its inverse relation $\mathcal R^{-1}$ defines (or induces) a mapping from the power set of $T$ to the power set of $S$:


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

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

Comment
Note that it is not necessarily the case that $f_\mathcal R^{-1} = f_{\mathcal R^{-1}}$.

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.