Image of 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 $\RR_2$.

Then the image of $\RR_2 \circ \RR_1$ is given by:

$\Img {\RR_2 \circ \RR_1} = \Img {\Img {\RR_1} \cap \Preimg {\RR_2} }$

Proof

We have by definition of composition of $\RR_1$ and $\RR_2$:

$\RR_2 \circ \RR_1 := \set {\tuple {x, z} \in S_1 \times T_2: \exists y \in S_2 \cap T_1: \tuple {x, y} \in \RR_1 \land \tuple {y, z} \in \RR_2}$

Let $\left({x, z}\right) \in \RR_2 \circ \RR_1$.

By definition of image:

$z \in \Img {\RR_2 \circ \RR_1}$

By definition of composition of $\RR_1$ and $\RR_2$:

$z \in \Img {\RR_2}$

But also:

$z \in \Img {\Img {\RR_1} }$

We also have that $\Img {\RR_2} = \Img {\Preimg {\RR_2} }$

That is: $z \in \Img {\Img {\RR_1} } \cap \Img {\Preimg {\RR_2} }$

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