∀ {α : Type u} {β : Type v} [inst : TopologicalSpace α] [inst_1 : TopologicalSpace β] {f : α → β},
  Topology.IsQuotientMap f →
    (∀ (y : β), IsConnected (f ⁻¹' {y})) → ∀ (a : α), f '' connectedComponent a = connectedComponent (f a)