Equivalence of Definitions of Dominate (Set Theory)

Theorem
Let $S, T$ be sets.

Definition 1 implies Definition 2
Let $f: S \to T$ be an injection.

By Injection to Image is Bijection, $f$ is a bijection from $S$ to the image of $f$.

Definition 2 implies Definition 1
Let $T' \subseteq T$ such that there exists a bijection $f: S \to T'$.

Let $i: T' \to T$ be the inclusion of $T'$ in $T$.

Then by Composite of Injections is Injection, $i \circ f: S \to T$ is an injection.