Image of Element under Composite Relation with Common Codomain and Domain

Theorem
Let $\RR_1 \subseteq S \times T$ and $\RR_2 \subseteq T \times U$ be relations.

Let $\RR_2 \circ \RR_1 \subseteq S \times U$ be the composition of $\RR_1$ and $\RR_2$.

Let $x \in S$.

Then:
 * $\RR_2 \sqbrk {\map {\RR_1} x} = \map {\paren{\RR_2 \circ \RR_1}} x$

Proof
We have:

Also see

 * Image of Subset under Composite Relation with Common Codomain and Domain


 * Image of Element under Composite Relation


 * Image of Subset under Composite Relation