Distance-Preserving Mapping is Injection of Metric Spaces

Theorem
Let $M_1 = \left({A_1, d_1}\right)$ and $M_2 = \left({A_2, d_2}\right)$ 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 $\phi \paren {a} = \phi \paren{b}$.

Then by the definition of a metric space:


 * $d_2 \tuple{ \phi \paren a, \phi \paren b } = 0$

By the definition of a distance-preserving mapping then:
 * $d_1 \tuple{ a, b } = 0$

Thus by the definition of a metric space:
 * $a = b$

Hence $\phi$ is injective.