Domain Topology Contains Initial Topology iff Mappings are Continuous

Theorem
Let $\struct{Y, \tau}$ be a topological space.

Let $\family {\struct{X_i, \tau_i}}_{i \mathop \in I}$ be a family of topological spaces.

Let $\family {f_i}_{i \mathop \in I}$ be a family of mappings $f_i : Y \to X_i$.

Let $\tau'$ be the initial topology on $Y$ with respect to $\family {f_i}_{i \mathop \in I}$.

Then:
 * $\tau' \subseteq \tau$ $\forall i \in I : f_i: \struct{Y, \tau} \to \struct{X_i, \tau_i}$ is continuous.