# Restriction of Homomorphism is Homomorphism

## Theorem

Let $\struct {S, \circ}$ and $\struct {T, \odot}$ be algebraic structures.

Let $\phi: S \to T$ be a homomorphism.

Let $A \subseteq S$ be a subset of $S$.

Then the restriction of $\phi$ to $A \times \Img A$ is also a homomorphism.

## Proof

 $\ds \forall x, y \in A: \,$ $\ds \map \phi {x \circ_A y}$ $=$ $\ds \map \phi {x \circ y}$ Definition of Restriction of Operation $\ds$ $=$ $\ds \map \phi x \odot \map \phi y$ as $x, y \in S$ $\ds$ $=$ $\ds \map \phi x \odot_{\Img A} \map \phi y$ Definition of Restriction of Operation

$\blacksquare$