Definition:Inverse Mapping

If the inverse $$f^{-1}$$ of a mapping $$f$$ is itself a mapping, then it is called the inverse mapping of $$f$$.

Thus, from the definition of a mapping, for $$f^{-1}$$ to be the inverse mapping of $$f$$:

$$\forall y \in T: \left({x_1, y}\right) \in f \land \left({x_2, y}\right) \in f \Longrightarrow x_1 = x_2$$

and

$$\forall y \in T: \exists x \in S: \left({x, y}\right) \in f$$

When $$f^{-1}$$ is a mapping, we say that "$$f$$ has an inverse mapping".