Image is a subgroup

Proposition (Image is a subgroup). Let $f:G\to H$ be a group homomorphism . Let $\mathrm{im}(f)$ be its image . Then $\mathrm{im}(f)$ is a subgroup of $H$.

Context. Together with “kernel is normal,” this gives the basic structural picture of any homomorphism: it factors through a quotient of $G$ onto a subgroup of $H$.