proof of Fodor’s lemma

If we let $f^{-1}:\kappa \to P(S)$ be the inverse of $f$ restricted to $S$ then Fodor’s lemma is equivalent to the claim that for any function such that $\alpha \in f(\kappa )\to \alpha >\kappa$ there is some $\alpha \in S$ such that $f^{-1}(\alpha )$ is stationary.

Then if Fodor’s lemma is false, for every $\alpha \in S$ there is some club set $C_{\alpha }$ such that $C_{\alpha }\cap f^{-1}(\alpha )=\mathrm{\varnothing }$. Let . The club sets are closed under diagonal intersection, so $C$ is also club and therefore there is some $\alpha \in S\cap C$. Then $\alpha \in C_{\beta }$ for each , and so there can be no such that $\alpha \in f^{-1}(\beta )$, so $f(\alpha )\ge \alpha$, a contradiction.