An ambiguity on the axiom of choice...

Enderton, in his book, makes an example for using the axiom of choice:

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For example, assume we are given a relation R. For any particular x in dom R, there exists some Y0 for which xRY0. And we can conclude that
for any x in dom R, there is a singleton {Y0}included in {t| xRt}. We have not yet used the axiom of choice. what does require the axiom is
saying that for any x in dom R, there is some Yx for which xRYx, and then putting all these Yx's together into a set, e.g.,
{Yx | x belongs to dom R}. This in effect is making many choices, one for each x in dom R, which may be an infinite set.

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Now my questions:

1- Isn't this set the same as ran R? If not, then what's the difference?
2- Which one of alternatives of the axiom of choice have we chosen for this problem? [Explain all of the process please]

A bit confused about this axiom!!

Thanks.

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Originally Posted by Mathelogician

1- Isn't this set the same as ran R? If not, then what's the difference?

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The set {Yx | x belongs to dom R} is a crosscut of the image of R. For each x it selects a single y such that xRy. There may be another y' such that xRy', but the set will contain y' only if x'Ry' for some x' ≠ x. In general, the image of R is greater than this set.

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Originally Posted by Mathelogician

2- Which one of alternatives of the axiom of choice have we chosen for this problem?

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We can use the standard formulation of the AC: For any set A of nonempty sets, there exists a choice function f defined on A. Suppose the domain of R is X and let R(x) = {t | xRt}. Let A = {R(x) | x ∈ X}. Then there exists a choice function f on A, i.e., f(R(x)) ∈ R(x) for every x ∈ X. The image of f is the set the book is talking about.

Thanks emarkov,
But why do you think Enderton is discussed this alternative of the axiom (using the choice function) as follows:
For any set A there exists a function F(a choice function for A) such that the domain of F is the set of nonempty subsets of A, and such that F(B)∈B for every nonempty subset B of A.
Does it agree with the usual one which talks about choosing elements from nonempty elements of A?

Wikipedia says that the following forms of the AC are equivalent:

(1) There exists a choice function on any family of nonempty sets;
(2) There exists a choice function on any powerset with the empty set removed.

Indeed, clearly, (2) is a special case of (1). Conversely, assume (2) and let A be a family of nonempty sets. Then build a choice function on the powerset of ⋃A using (2) and take its restriction to A.