A question about a proof of Hausdorff's Formula

Question

My textbook Introduction to Set Theory by Hrbacek and Jech presents Hausdorff's Formula:

and its corresponding proof:

I am unable to deduce 3. from 1. and 2. as stated in the proof.

1. Each function $f:\omega_\beta \to \omega_{\alpha+1}$ is bounded

2. The definition of ordinal exponentiation: $\omega_{\alpha+1}^{\omega_\beta}=\sup \{\omega_{\alpha+1}^{\lambda} \mid \lambda< \omega_\beta\}=\bigcup_{\lambda< \omega_\beta}\omega_{\alpha+1}^{\lambda}$

3. $\omega_{\alpha+1}^{\omega_\beta}=\bigcup_{\gamma < \omega_{\alpha+1}}\gamma^{\omega_\beta}$

Could you please elaborate on this point? Thank you for your help!

Answer 1

It is not ordinal exponentiation, it is cardinal exponentiation, i.e. $\omega_{\alpha+1}^{\omega_\beta}$ is the set of all functions $\omega_\beta\to \omega_{\alpha+1}.$ If every such function is bounded, then every such function is a function $\omega_\beta\to \gamma$ for some $\gamma < \omega_{\alpha}$ and hence $ \omega_{\alpha+1}^{\omega_\beta} = \bigcup_{\gamma < \omega_{\alpha+1}}\gamma^{\omega_\beta}.$