# Image of Union under Relation/Family of Sets

## Theorem

Let $S$ and $T$ be sets.

Let $\family {S_i}_{i \mathop \in I}$ be a family of subsets of $S$.

Let $\RR \subseteq S \times T$ be a relation.

Then:

$\ds \RR \sqbrk {\bigcup_{i \mathop \in I} S_i} = \bigcup_{i \mathop \in I} \RR \sqbrk {S_i}$

where $\ds \bigcup_{i \mathop \in I} S_i$ denotes the union of $\family {S_i}_{i \mathop \in I}$.

## Proof

 $\ds t$ $\in$ $\ds \RR \sqbrk {\bigcup_{i \mathop \in I} S_i}$ $\ds \leadstoandfrom \ \$ $\ds \exists s \in \bigcup_{i \mathop \in I} S_i: \,$ $\ds t$ $\in$ $\ds \map \RR s$ Image of Subset under Relation equals Union of Images of Elements $\ds \leadstoandfrom \ \$ $\ds \exists i \in I: \exists s \in S_i: \,$ $\ds t$ $\in$ $\ds \map \RR s$ Definition of Union of Family $\ds \leadstoandfrom \ \$ $\ds \exists i \in I: \,$ $\ds t$ $\in$ $\ds \RR \sqbrk {S_i}$ Definition of Image of Subset under Relation $\ds \leadstoandfrom \ \$ $\ds t$ $\in$ $\ds \bigcup_{i \mathop \in I} \RR \sqbrk {S_i}$ Definition of Union of Family

$\blacksquare$