Values of a Map between Classes, and Distances in a Metric Space

In the setting of The Real Numbers, with the Natural Numbers, Integers and Rationals Identified with Subsets of the Reals, and Completeness, let $A$ and $B$ be arbitrary classes introduced in this item, not necessarily sets, let $F$ be a class that is a map $F:A\to B$ in the sense of that definition, which applies to classes, and let $a\in A$; $F(a)$ is the value of $F$ at $a$, which is defined because $\operatorname{dom}F=A$. Unlike Values, Restriction and Equality of a Map, and the Map $u\mapsto t(u)$ as a Defined Set Symbol (ii), no hypothesis that $A$ or $B$ is a set is made.

(i) $F(a)$ is a set.

(ii) $(a,F(a))\in F$, where $(a,F(a))$ is the ordered pair.

(iii) $F(a)\in B$.

(iv) Let $(M,d)$ be a metric space, so that, by Metric Space, $M$ is a set and $d:M\times M\to\mathbb{R}$ is a map from the Cartesian product $M\times M$ to the real numbers, and let $x,y\in M$. Then $d(x,y)\in\mathbb{R}$, with $d(x,y)$ as in Sets and Maps: Ordinary Notation (v).