Order Isomorphism between Ordinals and Proper Class/Lemma

Lemma for Order Isomorphism between Ordinals and Proper Class
Suppose the following conditions are met:

Let $A$ be a class.

We allow $A$ to be a proper class or a set.

Let $\struct {A, \prec}$ be a strict well-ordering.

Let every $\prec$-initial segment be a set, not a proper class.

Let $\Img x$ denote the image of a subclass $x$.

Let $G$ equal the class of all ordered pairs $\tuple {x, y}$ satisfying:
 * $y \in A \setminus \Img x$
 * The initial segment $A_y$ of $\struct {A, \prec}$ is a subset of $\Img x$

Let $F$ be a mapping with a domain of $\On$.

Let $F$ also satisfy:
 * $\map F x = \map G {F \restriction x}$

Then:


 * $G$ is a mapping
 * $\map G x \in A \setminus \Img x \iff A \setminus \Img x \ne \O$

Note that only the first four conditions need hold: we may construct classes $F$ and $G$ satisfying the other conditions using transfinite recursion.

Proof
Therefore, we may conclude, that $G$ is a single-valued relation and therefore a mapping.

For the second part:

Furthermore:


 * $\map G x \in A \setminus \Img x \implies A \setminus \Img x \ne \O$ by the definition of non-empty.

Also see

 * Transfinite Recursion
 * Condition for Injective Mapping on Ordinals
 * Maximal Injective Mapping from Ordinals to a Set