Inverse of Bijection is Bijection

Theorem

Let $f: S \to T$ be a bijection in the sense that:

$(1): \quad f$ is an injection
$(2): \quad f$ is a surjection.

Then the inverse $f^{-1}$ of $f$ is itself a bijection by the same definition.

Proof

Let $f$ be both an injection and a surjection.

From Mapping is Injection and Surjection iff Inverse is Mapping it follows that its inverse $f^{-1}$ is a mapping.

$\paren {f^{-1} }^{-1} = f$

Thus the inverse of $f^{-1}$ is $f$.

But then $f$, being a bijection, is by definition a mapping.

So from Mapping is Injection and Surjection iff Inverse is Mapping it follows that $f^{-1}$ is a bijection.

$\blacksquare$