Image of Element under Composite Relation

Theorem
Let $\RR_1 \subseteq S_1 \times T_1$ and $\RR_2 \subseteq S_2 \times T_2$ be relations.

Let $\RR_2 \circ \RR_1 \subseteq S_1 \times T_2$ be the composition of $\RR_1$ and $\RR_2$.

Let $x \in S_1$.

Then:
 * $\RR_2 \sqbrk {\map {\RR_1} x \cap S_2} = \map {\paren{\RR_2 \circ \RR_1}} x$

Proof
We have:

Also see

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


 * Image of Subset under Composite Relation


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