That's correct. We have a collection, which we can think of as a set. It contains pairs of numbers. The only rule we impose on the elements of the set is that for any two pairs of numbers in the collection such that the first number is the same in both pairs, it must also be the case that the second number is the same. We can think of the first number in a pair as the x-coordinate, and the second number in a pair as the y-coordinate.

Assume we have two points (x1, y1) and (x1,y2). x1, y1, and y2 are variables. We don't know what their values are. All we know is that x1=x1, ie the x-coordinate of both points is the same. The rule says that the y-coordinates must be the same, ie y1=y2, ie we must be talking about the same point. There can be no two points with the same x-coordinate and a different y-coordinate.