Image of Element under Inverse Mapping

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

Let $f: S \to T$ be a mapping such that its inverse $f^{-1}: T \to S$ is also a mapping.

Then:
 * $\forall x \in S, y \in T: f \paren x = y \iff f^{-1} \paren y = x$

Sufficient Condition
Let $f: S \to T$ be a mapping.

From the definition of inverse mapping:
 * $f^{-1} = \set {\tuple {y, x}: \tuple {x, y} \in f}$

Let $y = f \paren x$.

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

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

However, $f^{-1}$ is a mapping.

Therefore, by definition:


 * $\forall y \in T: \tuple {y, x_1} \in f^{-1} \land \tuple {y, x_2} \in f^{-1} \implies x_1 = x_2$

Thus:
 * $\forall s \in f^{-1} \paren y: s = x$

Thus:
 * $f^{-1} \paren y = \set x$

That is:
 * $x = f^{-1} \paren y$

Necessary Condition
Let $f^{-1} \paren y = x$.

By definition of inverse mapping:
 * $f \paren x = y$