# Image of Union under Mapping/Proof 1

## Theorem

Let $S$ and $T$ be sets.

Let $f: S \to T$ be a mapping.

Let $A$ and $B$ be subsets of $S$.

Then:

$f \sqbrk {A \cup B} = f \sqbrk A \cup f \sqbrk B$

This can be expressed in the language and notation of direct image mappings as:

$\forall A, B \in \powerset S: \map {f^\to} {A \cup B} = \map {f^\to} A \cup \map {f^\to} B$

## Proof

First we have:

 $\displaystyle A$ $\subseteq$ $\displaystyle A \cup B$ Set is Subset of Union $\displaystyle \leadsto \ \$ $\displaystyle f \sqbrk A$ $\subseteq$ $\displaystyle f \sqbrk {A \cup B}$ Image of Subset under Mapping is Subset of Image

 $\displaystyle B$ $\subseteq$ $\displaystyle A \cup B$ Set is Subset of Union $\displaystyle \leadsto \ \$ $\displaystyle f \sqbrk B$ $\subseteq$ $\displaystyle f \sqbrk {A \cup B}$ Image of Subset under Mapping is Subset of Image

 $\displaystyle \leadsto \ \$ $\displaystyle f \sqbrk A \cup f \sqbrk B$ $\subseteq$ $\displaystyle f \sqbrk {A \cup B}$ Union is Smallest Superset

$\Box$

Then:

 $\displaystyle y$ $\in$ $\displaystyle f \sqbrk {A \cup B}$ $\displaystyle \leadsto \ \$ $\displaystyle \exists x \in A \cup B: y$ $=$ $\displaystyle \map f x$ Definition of Image of Subset under Relation $\displaystyle \leadsto \ \$ $\displaystyle \exists x: x \in A \lor x \in B: y$ $=$ $\displaystyle \map f x$ Definition of Set Union $\displaystyle \leadsto \ \$ $\displaystyle y$ $\in$ $\displaystyle f \sqbrk A \lor y \in f \sqbrk B$ Definition of Image of Subset under Relation $\displaystyle \leadsto \ \$ $\displaystyle y$ $\in$ $\displaystyle f \sqbrk A \cup f \sqbrk B$ Definition of Set Union $\displaystyle \leadsto \ \$ $\displaystyle f \sqbrk {A \cup B}$ $\subseteq$ $\displaystyle f \sqbrk A \cup f \sqbrk B$ Definition of Subset

$\Box$

Thus we have:

 $\displaystyle f \sqbrk A \cup f \sqbrk B$ $\subseteq$ $\displaystyle f \sqbrk {A \cup B}$ $\displaystyle f \sqbrk {A \cup B}$ $\subseteq$ $\displaystyle f \sqbrk A \cup f \sqbrk B$ $\displaystyle \leadsto \ \$ $\displaystyle f \sqbrk A \cup f \sqbrk B$ $=$ $\displaystyle f \sqbrk {A \cup B}$ Definition of Set Equality

$\blacksquare$