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.