Nope:
http://support.apple.com/kb/ht4108
In the notes it says "3. Apple AV adapters cannot provide video output when used with Lightning to 30-pin Adapter or Lightning to 30-pin Adapter (0.2 m) adapter."
Also there's a difference in what is spec'ed to be supported as these guys pointed out http://www.iclarified.com/24410/new-iphone-5-may-not-support-video-out-over-ligh tning-connector :
iPhone 5: TV and Video
● AirPlay Mirroring to Apple TV support at 720p
● AirPlay video streaming to Apple TV (3rd generation) at up to 1080p and Apple TV (2nd generation) at up to 720p
iPhone 4S: TV and Video
● AirPlay Mirroring to Apple TV support at 720p
● AirPlay video streaming to Apple TV (3rd generation) at up to 1080p and Apple TV (2nd generation) at up to 720p
● Video mirroring and video out support: Up to 1080p with Apple Digital AV Adapter or Apple VGA Adapter (adapters sold separately)
● Video out support at 576p and 480p with Apple Component AV Cable; 576i and 480i with Apple Composite AV Cable (cables sold separately).