# Relations such that each member of the range is the unique value of some member of the domain

A relation $$R$$ may have the property that each element $$y$$ of $$\mathrm{ran}\, R$$ is the unique right value of $$R$$ at some object, that is, for some object $$x$$, $$\langle x,z\rangle \in R$$ if and only if $$z = y$$. Is there a recognized name for this property?

One reason that there might be is that it is a necessary and sufficient condition for the direct image function $$R^{\rightarrow}\, \colon\, \mathcal{P}(\mathrm{dom}\, R) \to \mathcal{P}(\mathrm{ran}\, R)$$ to be a surjection; where, for each member $$A$$ of $$\mathcal{P}(\mathrm{dom}\, R)$$, $$R^{\rightarrow}(A)$$ is the direct image of $$A$$ under $$R$$, that is, the set of objects $$y$$ such that, for some member $$x$$ of $$A$$, $$\langle x, y \rangle \in R$$.

• Are you just looking for "is a function" (or some equivalent thereof), or am I misreading your question? Apr 3 at 1:47
• You're misreading the question. The property is strictly weaker than that of being a function. For example, {<0,0>, <1,0>,<1,2>,<2,2>} satisfies it but is not a function. Apr 3 at 15:20