Image of Linear Transformation is Submodule

Theorem
Let $\struct {R, +_R, \times_R}$ be a ring.

Let $\struct {G, +_G, \circ_G}_R$ and $\struct {H, +_H, \circ_H}_R$ be $R$-modules.

Let $\phi: G \to H$ be a linear transformation.

Let $\Img \phi$ denote the image set of $\phi$.

Then $\Img \phi$ is a submodule of $H$.

Proof
By Module is Submodule of Itself, $\struct {G, +_G, \circ_G}_R$ is a submodule of $\struct {G, +_G, \circ_G}_R$.

The result follows from Image of Submodule under Linear Transformation is Submodule.