Your car battery outputs between 11 and 12.5 volts when the car is not running.
In order for the alternator to charge the battery and run the electronics in the car it outputs a slightly higher voltage. So any time your engine is running, any lines that have 12V from the battery will likely read about 14 volts. Quick way to tell is with a voltmeter. Red lead to one of the bulb wires (when the bulb is on) and black lead to somewhere that has continuity to the frame of the vehicle.
Most automotive devices have a range from ~10 volts to ~16 volts for this reason.
I have not had any experience with the pro mini but i ran a Mega1280 to control some led's in my truck for a long while. I ended up using an old car power adapter that output 9Vdc and just used it as an input supply for the Mega to make sure i didn't have a heat issue on the board. Worked great until i scavenged the board for other project 