Distance-Preserving Mapping is Injection of Metric Spaces

Theorem

Let $M_1 = \struct {A_1, d_1}$ and $M_2 = \struct {A_2, d_2}$ be metric spaces.

Let $\phi: M_1 \to M_2$ be a distance-preserving mapping.

Then $\phi$ is an injection.

Proof

Let $a, b \in A_1$ and suppose that $\map \phi a = \map \phi b$.

Then by the definition of a metric space:

$\map {d_2} {\map \phi a, \map \phi b} = 0$

By the definition of a distance-preserving mapping then:

$\map {d_1} {a, b} = 0$

Thus by the definition of a metric space:

$a = b$

Hence $\phi$ is injective.

$\blacksquare$