Inverse Element of Injection

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Theorem

Let $S$ and $T$ be sets.

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


Then:

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


Proof

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:

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

$\Box$


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 Image of Preimage under Mapping: Corollary:

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

and so by definition of direct image mapping:

$y = \map f x$

$\blacksquare$


Sources