Can vtp function in a network where vlan1 has been shutdown?
You can shut down the vlan 1 L3 SVI (Switched Virtual Interface) and you can clear vlan 1 from all trunks but Cisco switches will still use vlan for most management traffic. I say most as DTP actually uses the native vlan which may or may not be vlan 1.
But CDP/PaGP/VTP all still use vlan 1 regardless of what you do on the switch. By shutting down vlan 1 interface and clearing off trunks you are effectively disabling vlan 1 from being used for end devices or to manage the switch but as i say this has no effect on the majority of Cisco management traffic.
Hence the reason VTP still works.