Characterization of Strictly Increasing Mapping on Woset

Lemma
Let $J$ and $E$ be well-ordered sets.

Let $h: J \to E$ be a mapping.

Let $S_\alpha$ denote an initial segment determined by $\alpha$.


 * $(1): \quad h$ is strictly increasing and its image is either all of $E$ or an initial segment of $E$


 * $(2): \quad \forall \alpha \in J: \map h \alpha = \map \min {E \setminus h \sqbrk {S_\alpha} }$, and $h \sqbrk {S_\alpha} = S_{\map h \alpha}$

where:


 * $h \sqbrk {S_\alpha}$ denotes the image of $S_\alpha$ under $h$


 * $\min$ denotes the smallest element of the set.

$(1)$ implies $(2)$
Suppose $h$ satisfies:


 * $h$ is strictly increasing and its image is either all of $E$ or an initial segment of $E$

Then for any $x,y \in J$:

$(2)$ implies $(1)$
Suppose $h$ satisfies:


 * $\map h \alpha = \map \min {E \setminus h \sqbrk {S_\alpha} }$

By the Principle of Recursive Definition for Well-Ordered Sets, $h$ is thus uniquely determined.

Then:

Thus for every $x \in S_y$, we have that $\map h x \in S_{\map h y}$.

Therefore $h$ is an strictly increasing mapping.

Furthermore, the image set of $h$ is the union of initial segments in $E$.

By Union of Initial Segments is Initial Segment or All of Woset, $h \sqbrk J$ is an initial segment of $E$ or all of $E$.