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 $\family {S_i}_{i \mathop \in I}$.

$\ds t$

$\in$

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

$\ds \leadstoandfrom \ \ $

$\ds \exists s \in \bigcup_{i \mathop \in I} S_i: \, $

$\ds t$

$\in$

$\ds \map \RR s$

$\ds \leadstoandfrom \ \ $

$\ds \exists i \in I: \exists s \in S_i: \, $

$\ds t$

$\in$

$\ds \map \RR s$

$\ds \leadstoandfrom \ \ $

$\ds \exists i \in I: \, $

$\ds t$

$\in$

$\ds \RR \sqbrk {S_i}$

$\ds \leadstoandfrom \ \ $

$\ds t$

$\in$

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

$\blacksquare$

1955: John L. Kelley: General Topology ... (previous) ... (next): Chapter $0$: Relations: Theorem $5 \ \text{(d)}$
1971: Robert H. Kasriel: Undergraduate Topology ... (previous) ... (next): $\S 1.12$: Set Inclusions for Image and Inverse Image Sets: Theorem $12.5 \ \text{(a)}$