Characterization for Topological Evaluation Mapping to be Embedding/Sufficient Condition

Theorem
Let $X$ be a topological space.

Let $\family {Y_i}_{i \mathop \in I}$ be an indexed family of topological spaces for some indexing set $I$.

Let $\family {f_i : X \to Y_i}_{i \mathop \in I}$ be an indexed family of continuous mappings.

Let $\ds Y = \prod_{i \mathop \in I} Y_i$ be the product space of $\family {Y_i}_{i \mathop \in I}$.

Let $f : X \to Y$ be the evaluation mapping defined by:
 * $\forall x \in X : \map f x = \family{\map {f_i} x}_{i \mathop \in I}$

Let:
 * $(1)\quad$ the topology on $X$ be the initial topology with respect to $\family {f_i}_{i \mathop \in I}$
 * $(2)\quad$ the family $\family {f_i}$ separate points

Then:
 * $f$ is an embedding

Proof
From User:Leigh.Samphier/Topology/Evaluation Map is Injective iff Mappings Separate Points:
 * $f$ is an injection

From Injection to Image is Bijection:
 * $f \restriction_{f \sqbrk X} : X \to f \sqbrk X$ is a bijection

From User:Leigh.Samphier/Topology/Every Topological Evaluation Mapping is Continuous:
 * $f$ is continuous

From Continuity of Composite with Inclusion on Mapping:
 * $f \restriction_{f \sqbrk X}$ is continuous