Chain conditions preserve high cofinalities and ccc preserves cardinals

If $p⊩\dot{f}:\check{\mu }\to \check{\lambda }$, choose for each $\xi <\mu$ a maximal antichain below $p$ deciding $\dot{f}(\xi )$. Each has size $<\theta$, so the ground-model set $B_{\xi }$ of possible values has size $<\theta$. Then $p⊩\mathrm{ran}(\dot{f})\subseteq {\bigcup }_{\xi <\mu }{B}_{\xi }$. AC is used for maximal antichains and their simultaneous selection.

Let $\lambda \ge \theta$ be regular and $\mu <\lambda$. If a condition forced $\dot{f}:\mu \to \lambda$ cofinal, step 1.1 and regularity would put its range inside a ground set of size $\max (\mu ,<\theta )<\lambda$, which is bounded in $\lambda$, contradiction. Thus regular cofinalities at least $\theta$ are preserved; F2 transfers this to every ground cofinality at least $\theta$.

Suppose a ground cardinal $\lambda \ge \theta$ were collapsed. By F4, some $\mu <\lambda$ and a condition $p$ would force a surjection $\dot{f}:\mu \to \lambda$. Step 1.1 puts its range inside the ground set $U={\bigcup }_{\xi <\mu }{B}_{\xi }$, with $\mathrm{\mid }{B}_{\xi }\mathrm{\mid }<\theta$. If $\lambda =\theta$, regularity of $\theta$ gives $\mathrm{\mid }U\mathrm{\mid }<\theta$; if $\lambda >\theta$, cardinal arithmetic under AC gives $\mathrm{\mid }U\mathrm{\mid }\le \mu \cdot \theta =\max (\mu ,\theta )<\lambda$ (with the finite cases immediate). Either way $p$ cannot force $\dot{f}$ onto $\lambda$. Thus every ground cardinal at least $\theta$ remains a cardinal. For ccc, $\theta ={\mathrm{\aleph }}_1$; finite and countable cardinals and cofinalities are absolute, so steps 2.1 and 3.1 cover all of them.