Neil Trevett, the current president of Khronos Group and a vice president at NVIDIA, made an on-the-record statement to acknowledge the start of the Vulkan API. The quote came to me via Ryan, but I think it is a copy-paste of an email, so it should be verbatim.
Many companies have made great contributions to Vulkan, including AMD who contributed Mantle. Being able to start with the Mantle design definitely helped us get rolling quickly – but there has been a lot of design iteration, not the least making sure that Vulkan can run across many different GPU architectures. Vulkan is definitely a working group design now.
So in short, the Vulkan API was definitely started with Mantle and grew from there as more stakeholders added their opinion. Vulkan is obviously different than Mantle in significant ways now, such as its use of SPIR-V for its shading language (rather than HLSL). To see a bit more information, check out our article on the announcement.
Update: AMD has released a statement independently, but related to Mantle's role in Vulkan