Image of Subset 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 $A \subseteq S_1$.

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

Proof
We have:

Also see

 * Image of Element under Composite Relation


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


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