Choice preservation by well-ordered names ($M\models \mathrm{AC}⟹M[G]\models \mathrm{AC}$)

If the ground model has axiom of choice, well-order the set of subnames appearing in a forcing name for an arbitrary extension set. Evaluate those activated by the generic filter. Each element of the extension set has a least preimage index; ordering by these indices well-orders the set. Only ground choice and the ZF part of the forcing theorem are needed.