Homeomorphic Image of Sub-Basis is Sub-Basis

Theorem
Let $T_\alpha = \struct{S_\alpha, \tau_\alpha}$ and $T_\beta = \struct{S_\beta, \tau_\beta}$ be topological spaces.

Let $\SS \subseteq \tau_\alpha$ be a sub-basis for $\tau_\alpha$.

Let $\phi: T_\alpha \to T_\beta$ be a homeomorphism.

Then:
 * $\SS' = \set{\phi \sqbrk S : S \in \SS}$ is a sub-basis for $\tau_\beta$

Proof
By definition of homeomorphism:
 * $\forall U \subseteq X_\alpha : U \in \tau_\alpha \iff \phi \sqbrk U \in \tau_\beta$

By definition of sub-basis:
 * $\SS \subseteq \tau_\alpha$

Hence:
 * $\SS'\subseteq \tau_\beta$