As yet I do not think there is any documentation found to prove that Anthony Jansen van Sallee was truly a son of Jan Jansen van Haarlem. There has been lots of "family lore" and stories around this but as yet I have not seen anything to actually prove this. If someone knows of a document or something else that might prove this link I would be most interested.
In that Anthony was in New Amsterdamthese early records were well preserved and are proof of his descendants. Who Anthony's father was though is another matter.