Can a Person Be Convicted of Threatening Harm Online?

Is it reasonable to judge someone for using interstate communications to threaten harm online?

Anthony, distraught after he lost his job and his wife left him, takes his frustration onto Facebook. Under the assumed name, Anthony posts rap lyrics telling his wife, “There’s one way to love ya but a thousand ways to kill ya,” adding, “I’m not going to rest until your body is a mess, soaked in blood and dying from all the little cuts.” He also writes, regarding a federal law enforcement agent who had questioned him, “Pull my knife, flick my wrist, and slit her throat.” Anthony is eventually arrested for making these online statements. Does the prosecution have to prove that Anthony intended to threaten someone?