Preimage of Mapping equals Domain

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

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

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

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

The result follows from the definition of set equality.