Mapping is Injection and Surjection iff Inverse is Mapping

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Theorem

Let $S$ and $T$ be sets.

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

Then:

$f: S \to T$ can be defined as a bijection in the sense that:
$(1): \quad f$ is an injection
$(2): \quad f$ is a surjection

if and only if:

the inverse $f^{-1}$ of $f$ is itself a mapping.


Proof

Necessary Condition

Let $S$ and $T$ be sets.

Let $f: S \to T$ be a mapping such that:

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


Then the inverse $f^{-1}$ of $f$ is itself a mapping.


Sufficient Condition

Let $S$ and $T$ be sets.

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

Let the inverse $f^{-1} \subseteq T \times S$ itself be a mapping.


Then:

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


Sources