∀ {α : Type u_1} [inst : DecidableEq α] [inst_1 : Fintype α] {β : Type u_2} [inst_2 : DecidableEq β]
  [inst_3 : Fintype β] {p : β → Prop} [inst_4 : DecidablePred p] (f : α ≃ Subtype p) {g : Equiv.Perm α},
  (g.extendDomain f).support = Finset.map f.asEmbedding g.support