Mac Book Pro 2013 connect to projector with resolution bigger then laptop screen
According to everymac.com, your particular MacBook Pro will display 2560x1600 maximum resolution (Thunderbolt) and 1920x1200 via HDMI.



If your projector supports 1920x1080, and it being "Full HD" as you described, it will most likey have an HDMI port.  All you will need is a quality mini Display Port (mdP) to HDMI adapter