Transfinite Recursion Theorem/Formulation 2/Lemma

Lemma for Transfinite Recursion Theorem: Formulation $2$

Let $\On$ denote the class of all ordinals.

Let $S$ denote the class of all ordinal sequences.

Let $g$ be a mapping such that $S \subseteq \Dom g$.

There exists an extending operation $E$ such that for every mapping $F$ on $\On$:

$\forall \alpha \in \On: \map {\paren {\map E {F \restriction \alpha} } } \alpha = \map g {F \restriction \alpha}$

Let $\theta_\alpha$ be an arbitrary ordinal sequence of length $\alpha$.

Let $\map E {\theta_\alpha}$ be the extending operation defined as:

$\map E {\theta_\alpha} := \theta_\alpha \cup \set {\tuple {\alpha, \map g {\theta_\alpha} } }$

Thus $\map E {\theta_\alpha}$ is an ordinal sequence of length $\alpha^+$.

Also, for $\beta \in \On$ such that $\beta < \alpha$:

$\map {\paren {\map E {\theta_\alpha} } } \beta = \map {\theta_\alpha} \beta$

but:

$\map {\paren {\map E {\theta_\alpha} } } \alpha = \map g {\theta_\alpha}$

Then for any mapping $F$ on $\On$:

$F \restriction \alpha$ is an ordinal sequence of length $\alpha$.

Hence:

$\map {\paren {\map E {F \restriction \alpha} } } \alpha = \map g {F \restriction \alpha}$

$\blacksquare$

2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $6$: Order Isomorphism and Transfinite Recursion: $\S 5$ Transfinite recursion theorems: Lemma $5.4$