Prikry forcing preserves every cardinal

Every ground cardinal $\lambda <\kappa$ remains a cardinal. Otherwise in $M[G]$ some ordinal $\alpha <\lambda$ would be bijective with $\lambda$, by F7. In $M$, cardinal absorption supplies a bijection coding $\alpha \times \lambda$ by an ordinal $\rho <\kappa$ (the finite cases are immediate and the infinite case uses F6). The graph of the alleged bijection therefore codes a new subset of $\rho$, but F1 says that subset, and hence the graph, belongs to $M$. This contradicts that $\lambda$ was a ground cardinal. Choice is used only through the ground cardinal comparisons and coding already stated in F6 and F7.

The ordinal $\kappa$ also remains a cardinal. If it were equinumerous in $M[G]$ with some $\alpha <\kappa$, let $\mu =\mathrm{\mid }\alpha {\mathrm{\mid }}^M<\kappa$ and obtain an injection $e:\kappa \to \mu$. Since F3 makes $\kappa$ a limit cardinal, the ground successor cardinal ${\mu }^{+}$ is still below $\kappa$; it remains a cardinal by step 1.1. Restricting $e$ to ${\mu }^{+}$ would inject that preserved successor cardinal into $\mu$, contradicting the defining minimality in F7.

Write the infinite cardinal $\kappa$ as ${\mathrm{\aleph }}_{\alpha }$ using F5. Then ${\kappa }^{+}={\mathrm{\aleph }}_{\alpha +1}$ by the successor convention in F7, and F5 makes ${\kappa }^{+}$ regular under Choice. F2 and F4 now imply that every ground cardinal at least ${\kappa }^{+}$ is preserved. Together with steps 1.1 and 2.1, this covers every ground cardinal.

Cardinal preservation is not cofinality preservation at $\kappa$: F8 supplies in $M[G]$ a cofinal map from $\omega$ into the still-cardinal ordinal $\kappa$, and proves $\mathrm{cf}(\kappa )=\omega$. Thus the two promised conclusions coexist without treating the cofinality change as a collapse.