Song of Ice and Fire Series Possible Ending

Okay, first things first, this is a theory I just had please don't cyber yell at me if you don't agree. This is largely contingent upon R+L=J and it is a prediction of the end of the series.

!Is the Song of Ice and Fire, being the title of the book series, referring to Jon Snow's right to the Iron Throne?

!When Dany is in the House of the Undying she see's Rhaegar Targaryen with Elia Martell while she is nursing a baby, she says that he is a "prince that was promised, and his is a song of ice and fire." Was that Rhaegar and Lyanna's baby, aka Jon Snow. Being a child concevied by a Stark from the frozen tundra that is Winterfell, and a three headed, fire breathing, dragon that is a Targaryen would mean he is a product of ice and fire.

Is there anything else in the books that would support this theory? Or is there something to completely shut it down? Again, if you think it's dumb please don't be rude about it.