Inverse Element of Injection

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

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

Then:


 * $\map {f^{-1} } y = x \iff \map f x = y$

Necessary Condition
Let $y = \map f x$.

From the definition of the preimage of an element:
 * $\map {f^{-1} } y = \set {x \in S: \tuple {y, x} \in f}$

Thus:
 * $x \in \map {f^{-1} } y$

By definition of injection, $\map {f^{-1} } y$ is a singleton:
 * $\map {f^{-1} } y = \set x$

which can be expressed as:
 * $\map {f^{-1} } y = x$

Sufficient Condition
Let $\map {f^{-1} } y = x$.

Thus by definition of direct image mapping:
 * $\map {f^\gets} {\set y} = \set x$

Then:
 * $\map {f^\to} {\map {f^\gets} {\set y} } = \map {f^\to} {\set x}$

So from the corollary to Image of Preimage under Mapping:


 * $\set y = \map {f^\to} {\set x}$

and so by definition of direct image mapping:


 * $y = \map f x$