Electronic device that is open for 6V and passes current once it reaches 12v?
A simple solution is to use a 5V relay, and put a zener with about 7V breakdown voltage in series with the coil. Something like 1N5342 or 1N5921 (6.8V) would be appropriate. This way, the relay will see 5.2V when there is 12V, and nothing when the voltage is only 6V (or anything under the breakdown voltage). This is a very simple circuit, which doesn't rely on some unspecified behavior.

Just put the zener the right way (cathode towards the positive supply). And size the relay appropriately (coil rated for 5VDC, contact specified for at least 20VDC or more, and check max amp capabilities against your projector's requirements). Better use some automotive-grade relay, by the way.

Schematic:







simulate this circuit &ndash; Schematic created using CircuitLab