What is the Definition of a Relation in Set Theory?

[Mr Davis 97]

I have an exercise in my set theory book that states the following: Show that a set $A$ is a relation iff $A \subseteq \operatorname{dom} A\times \operatorname{ran} B$.

This is an easy exercise, so I am not asking how to prove it. However, I am confused about one thing.

The forward direction is trivial, by the definition of the relation. However, the other direction confuses me, because it uses the dom and ran operations, which I thought were only defined for relations, when in fact we can't assume that $A$ is a relation. Is this saying that dom and ran are defined with respect to arbitrary sets? For example, if $A = \{1, (2,3)\}$, then dom A = 2 and ran A = 3, where the 1 is just disregarded?

 

[FactChecker]

What definition of "relation" do you have? The thing you are asked to prove to be equivalent is what I am used to seeing as the definition.
I would assume that "dom" and "ran" were just inserted to help you understand what roles A and B have, respectively.