Set-sized Easton realization on regular cardinals

Fix $\kappa \in \mathrm{dom}(F)$. Then $\kappa$ is an infinite regular cardinal, $F(\kappa )$ is a ground cardinal with $F(\kappa )>\kappa$ and $\mathrm{cf}(F(\kappa ))>\kappa$ by [F2], and by [F1] the models $M\subseteq M[G]$ have the same ordinals, cofinality function and cardinals, so $\kappa$ keeps its cofinality and $F(\kappa )$ is still a cardinal of $M[G]$.

The head and the tail are the factors of $P(F)$ at $\kappa$: $P(F)\cong P^{\le \kappa }\times P^{>\kappa }$ with $P^{>\kappa }$ ${\kappa }^{+}$-closed, $P^{\le \kappa }$ ${\kappa }^{+}$-cc and both set-sized [F3, F4]. The coordinate projections $G^{>\kappa }$ and $G^{\le \kappa }$ of $G$ are $M$-generic: if $D\subseteq P^{\le \kappa }$ is dense and $D\in M$, then $\{p\in P(F):p^{\le \kappa }\in D\}$ is dense in $P(F)$ and lies in $M$, so $G$ meets it and $G^{\le \kappa }$ meets $D$, and symmetrically for the tail; hence $M[G]=M[G^{>\kappa }][G^{\le \kappa }]$.

$(2^{\kappa }{)}^{M[G]}\le F(\kappa )$: let $x\in M[G]$ with $x\subseteq \kappa$. Its characteristic function is a function $\kappa \to M$ in $M[G]$, so $x\in M[G^{\le \kappa }]$ by [F6] applied to the pair $P^{>\kappa }$, $P^{\le \kappa }$ of step 1.2. By [F9] there is a $P^{\le \kappa }$-name $\sigma \in M$ with ${\sigma }_{G^{\le \kappa }}=x$. Form in $M$ the usual name ${\sigma }^{\mathrm{\prime }}$ for $\sigma \cap \check{\kappa }$; the top condition forces ${\sigma }^{\mathrm{\prime }}\subseteq \check{\kappa }$, and ${\sigma }_{G^{\le \kappa }}^{\mathrm{\prime }}=x$. By [F8] there is a nice $P^{\le \kappa }$-name $\tau \in M$ with ${\tau }_{G^{\le \kappa }}=x$. By [F7] the set of nice $P^{\le \kappa }$-names for subsets of $\kappa$ has at most $F(\kappa )$ elements in $M$, and $F(\kappa )$ is a cardinal of $M[G]$ by step 1.1, so the assignment $x\mapsto$ the ground well-order-least such $\tau$, which is defined in $M[G]$ using the well-order that [F10] gives in $M$, is an injection of the subsets of $\kappa$ in $M[G]$ into $F(\kappa )$. Hence $(2^{\kappa }{)}^{M[G]}\le F(\kappa )$.

$(2^{\kappa }{)}^{M[G]}\ge F(\kappa )$: for each $\beta <F(\kappa )$ form the $P^{\le \kappa }$-name ${\dot{x}}_{\beta }=\{⟨\check{\alpha },p⟩:\alpha <\kappa ,p\in P^{\le \kappa },p(\kappa ,\alpha ,\beta )=1\}$. Its value is a subset of $\kappa$. For each $\alpha <\kappa$, the head conditions deciding the coordinate $(\kappa ,\alpha ,\beta )$ are dense, so $\alpha$ belongs to $x_{\beta }:={\mathrm{val}}_{G^{\le \kappa }}({\dot{x}}_{\beta })$ exactly when the generic column at $(\alpha ,\beta )$ has bit $1$. For distinct $\beta ,{\beta }^{\mathrm{\prime }}$ and any head condition $p\in P^{\le \kappa }$, choose $\alpha <\kappa$ not occurring in any triple of $\mathrm{dom}(p)$, possible since $\mid \mathrm{dom}(p)\mid <\kappa$. Then $q=p\cup \{(\kappa ,\alpha ,\beta )\mapsto 1,(\kappa ,\alpha ,{\beta }^{\mathrm{\prime }})\mapsto 0\}$ is a head condition stronger than $p$ with $\mid \mathrm{dom}(q)\mid <\kappa$; adding these two coordinates also leaves the support bounds below $\kappa$ intact. The condition $q$ forces $\check{\alpha }\in {\dot{x}}_{\beta }$ and $\check{\alpha }\mathrm{\notin }{\dot{x}}_{{\beta }^{\mathrm{\prime }}}$. Thus the head conditions forcing ${\dot{x}}_{\beta }\mathrm{\ne }{\dot{x}}_{{\beta }^{\mathrm{\prime }}}$ are dense. By [F9] the map $\beta \mapsto x_{\beta }$ is an injection of $F(\kappa )$ into $\mathcal{P}(\kappa )$ in $M[G]$, and $(2^{\kappa }{)}^{M[G]}\ge F(\kappa )$.

Steps 2.1 and 2.2 and the fact that $F(\kappa )$ is a cardinal of $M[G]$ give $(2^{\kappa }{)}^{M[G]}=F(\kappa )$ for the fixed $\kappa$, and $\kappa \in \mathrm{dom}(F)$ was arbitrary, so the continuum function on $\mathrm{dom}(F)$ is realized; step 1.1 gives the preservation clause (a). This is the statement. ∎