Equivalence Class of Isometries is Subset of Equivalence Class of Homeomorphisms

Theorem
Let $\CC_I$ be an equivalence class of isometries on the set of metric spaces.

Then $\CC_I$ is a subset of an equivalence class of homeomorphisms on the set of metric spaces.

Proof
From Isometry of Metric Spaces is Equivalence Relation, all isometries can be partitioned into equivalence classes.

From Homeomorphism of Metric Spaces is Equivalence Relation, all homeomorphisms can be partitioned into equivalence classes.

Let $\CC_I$ be an equivalence class of isometries.

Let $f: M_1 \to M_2$ be an element of $\CC_I$ where $M_1$ and $M_2$ are some metric spaces.

From Isometry of Metric Spaces is Homeomorphism, $\CC_I$ is a homeomorphism.

Thus $\CC_I$ is an element of some equivalence class of homeomorphisms on the set of metric spaces.