Inverse Element of Injection

Theorem
Let $S$ and $T$ be sets.

Let $f: S \to T$ be an injection.

Then:


 * $f^{-1} \left({y}\right) = x \iff f \left({x}\right) = y$

Necessary Condition
Let $y = f \left({x}\right)$.

From the definition of the preimage of an element:
 * $f^{-1} \left({y}\right) = \left\{{x \in S: \left({y, x}\right) \in f}\right\}$

Thus:
 * $x \in f^{-1} \left({y}\right)$

By definition of injection, $f^{-1} \left({y}\right)$ is a singleton:
 * $f^{-1} \left({y}\right) = \left\{{x}\right\}$

which can be expressed as:
 * $f^{-1} \left({y}\right) = x$

Sufficient Condition
Let $f^{-1} \left({y}\right) = x$.

Thus by definition of induced mapping:
 * $f^\gets \left({\left\{{y}\right\} }\right) = \left\{{x}\right\}$

Then:
 * $f^\to \left({f^\gets \left({\left\{{y}\right\} }\right)}\right) = f^\to \left({\left\{{x}\right\}}\right)$

So from the corollary to Image of Preimage of Mapping:


 * $\left\{{y}\right\} = f^\to \left({\left\{{x}\right\}}\right)$

and so by definition of induced mapping:


 * $y = f \left({x}\right)$