Field Homomorphism Preserves Subfields/Corollary

Corollary to Field Homomorphism Preserves Subfields
Let $\left({F_1, +_1, \circ_1}\right)$ and $\left({F_2, +_2, \circ_2}\right)$ be fields.

The image of a field homomorphism $\phi: F_1 \to F_2$ is a subfield of $F_2$.

Proof
From Field is Subfield of Itself, $F_1$ is a subfield of $F_1$.

The result then follows directly from Field Homomorphism Preserves Subfields.