Preimage of Mapping equals Domain

Theorem
The preimage of a mapping is the same set as its domain:


 * $$\operatorname{Im}^{-1} \left({f}\right) = \operatorname{Dom} \left({f}\right)$$.

Proof
Let $$f \subseteq S \times T$$ be a mapping. Then:

$$ $$ $$

From Preimage Subset of Domain, we have that $$\operatorname{Im}^{-1} \left({f}\right) \subseteq S$$.

The result follows from the definition of set equality.