Correspondence between Set and Ordinate of Cartesian Product is Mapping

Theorem
Let $S$ and $T$ be sets such that $T \ne \O$.

Let $S \times T$ denote their cartesian product.

Let $t \in T$ be given.

Let $j_t \subseteq S \times \paren {S \times T}$ be the relation on $S \times {S \times T}$ defined as:


 * $\forall s \in S: \map {j_t} s = \tuple {s, t}$

Then $j_t$ is a mapping.

Proof
First it is to be shown that $j_t$ is left-total.

This follows from the fact that $j_t$ is defined for all $s$:
 * $\map {j_t} s = \tuple {s, t}$

Next it is to be shown that $j_t$ is many-to-one, that is:
 * $\forall s_1, s_2 \in S: \map {j_t} {s_1} \ne \map {j_t} {s_2} \implies s_1 \ne s_2$

We have that:

Hence the result.

Also see

 * Definition:Canonical Injection (Abstract Algebra) for an instance of this construct in the context of algebraic structures