Continuous Mapping on Finer Domain and Coarser Codomain Topologies is Continuous

Theorem
Let $\struct {X, \tau_1}$ and $\struct {Y, \tau_2}$ be topological spaces.

Let $f: \struct {X, \tau_1} \to \struct {Y, \tau_2}$ be a continuous mapping.

Let $\tau'_1$ be a finer topology on $X$ than $\tau_1$, that is, $\tau_1 \subseteq \tau'_1$.

Let $\tau'_2$ be a coarser topology on $Y$ than $\tau_2$, that is, $\tau'_2 \subseteq \tau_2$.

Then:
 * $f: \struct {X, \tau'_1} \to \struct {Y, \tau'_2}$ is a continuous mapping.

Proof
Let $U \in \tau'_2$.

Since $\tau'_2$ is a coarser topology than $\tau_2$:
 * $U \in \tau_2$

By definition of continuity:
 * $\map {f^{-1}} U \in \tau_1$

Since $\tau'_1$ is a finer topology than $\tau_1$:
 * $\map {f^{-1}} U \in \tau'_1$

Since $U$ was arbitrary, by definition of continuity:
 * $f: \struct {X, \tau'_1} \to \struct {Y, \tau'_2}$ is a continuous mapping