Inverse is Mapping implies Mapping is Injection and Surjection

Theorem

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.

Proof

This is divided into two parts:

Inverse is Mapping implies Mapping is Injection

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 $f$ is an injection.

Inverse is Mapping implies Mapping is Surjection

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 $f$ is a surjection.