Final Topology Contains Codomain 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 $f_i : X_i \to Y$.

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

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

Necessary Condition
Let $\tau \subseteq \tau'$.

From Equivalence of Definitions of Final Topology:
 * for each $i \in I$, $f_i: \struct{X_i, \tau_i} \to \struct{Y, \tau'}$ is $\tuple{\tau_i, \tau'}$-continuous

From Continuous Mapping on Finer Domain and Coarser Codomain Topologies is Continuous:
 * for each $i \in I$, $f_i: \struct{X_i, \tau_i} \to \struct{Y, \tau}$ is $\tuple{\tau_i, \tau}$-continuous

Sufficient Condition
For all $i \in I$, let $f_i: \struct{X_i, \tau_i} \to \struct{Y, \tau}$ be $\tuple{\tau, \tau_i}$-continuous.

From Equivalence of Definitions of Final Topology:
 * $\tau'$ is the finest topology on $Y$ such that each $f_i: \struct{X_i, \tau_i} \to \struct{Y, \tau'}$ is $\tuple{\tau_i, \tau'}$-continuous.

Then:
 * $\tau \subseteq \tau'$