Image of Element under Cartesian Product of Subsets

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Theorem

Let $S$ and $T$ be sets.

Let $A \subseteq S$ and $B \subseteq T$.

Let $\RR$ be the relation defined by the Cartesian product $A \times B$.

Then:

$\forall x \in A: \map \RR x = B$

Proof

We have:

\(\ds \forall x \in A: \, \)

\(\ds \map \RR s\)

\(=\)

\(\ds \set {t \in T: \tuple {s, t} \in \RR}\)

Definition of Image of Element under Relation

\(\ds \)

\(=\)

\(\ds \set {t \in T: \tuple {s, t} \in A \times B}\)

Definition of $\RR$

\(\ds \)

\(=\)

\(\ds \set {t \in T: s \in A, t \in B}\)

Definition of Cartesian Product

\(\ds \)

\(=\)

\(\ds \set {t \in T: t \in B}\)

as $s \in A$

\(\ds \)

\(=\)

\(\ds B\)

as $B \subseteq T$

$\blacksquare$