Restriction of Homeomorphism is Homeomorphism

Theorem
Let $T_1 = (S_1, \tau_1)$, $T_2 = (S_2, \tau_2)$ be topological spaces.

Let $f: S_1 \to S_2$ be a homeomorphism between $T_1$ and $T_2$.

Let $S$ be a subset of $S_1$.

Let $f \restriction_{S \times f(S)} : S \to f(S)$ be the restriction of $f$ to $S \times f(S)$.

Then $f \restriction_{S \times f(S)}$ is a homeomorphism between $S$ and $f(S)$, where both $S$ and $f(S)$ bear the subspace topology.

Proof
Since the restriction of a continuous mapping is continuous, $f \restriction_{S \times f(S)}$ is continuous with respect to the respective subspace topologies.

Since the inverse of a restriction is the restriction of the inverse, $(f \restriction_{S \times f(S)})^{-1}$ is well-defined and equal to $f^{-1} \restriction_{f(S) \times S}$.

Since the restriction of a continuous mapping is continuous, $f^{-1} \restriction_{f(S) \times S}$ is continuous with respect to the respective subspace topologies.

Since $(f \restriction_{S \times f(S)})^{-1} = f^{-1} \restriction_{f(S) \times S}$, $(f \restriction_{S \times f(S)})^{-1}$ is continuous.

By definition of homeomorphism, $f \restriction_{S \times f(S)}$ is a homeomorphism.