A two-coordinate Easton pattern

The domain $\{{\mathrm{\aleph }}_0,{\mathrm{\aleph }}_1\}$ is a set of infinite regular cardinals by [F2], the values $F({\mathrm{\aleph }}_0)=F({\mathrm{\aleph }}_1)={\mathrm{\aleph }}_3$ are cardinals by [F4], and $F$ is nondecreasing because its two values are equal.

$\mathrm{cf}(F(\kappa ))>\kappa$ holds at both domain points: $\mathrm{cf}({\mathrm{\aleph }}_3)={\mathrm{\aleph }}_3$, since ${\mathrm{\aleph }}_3$ is regular by [F2] and [F4], and ${\mathrm{\aleph }}_3>{\mathrm{\aleph }}_1>{\mathrm{\aleph }}_0$ by [F4].

Steps 1.1 and 1.2 verify all four clauses of [F1], so $F$ is a set-sized Easton function; the hypothesis of [F3] is met by the given ground model $M$ of ZFC+GCH and by the $M$-generic $G$.

Applying [F3] at the two domain points gives $(2^{{\mathrm{\aleph }}_0}{)}^{M[G]}=F({\mathrm{\aleph }}_0)={\mathrm{\aleph }}_3$ and $(2^{{\mathrm{\aleph }}_1}{)}^{M[G]}=F({\mathrm{\aleph }}_1)={\mathrm{\aleph }}_3$, while clause (a) of [F3] gives that $M[G]$ has the same ordinals, cofinalities and cardinals as $M$.

The Axiom of Choice is used in the regularity of ${\mathrm{\aleph }}_1$ and ${\mathrm{\aleph }}_3$ of steps 1.1 and 1.2 and is part of the ZFC ground model on which [F3] is stated; no value at a singular cardinal is asserted, since $F$ is only defined on $\{{\mathrm{\aleph }}_0,{\mathrm{\aleph }}_1\}$. The displayed equalities are exactly the statement. ∎