Is it correct to say that $P=NP$ implies $P=NPC$?
I was reviewing the definition of NP-complete and I noticed this diagram which states that if $P=NP$, then $P=NP=NPC$. However, it seems to me that if $P=NP$, then $NPC=NP \setminus \{\mathbb{N},\emptyset\}$, because there can't be a reduction from a problem with two possible solutions to a problem with only one possible solution, as seems to be required by the definitions. So which statement is correct? Is this a case of loose terminology similar to the issue of whether or not zero should be considered a "natural number"? Or maybe the diagram is right and there is something else wrong with my understanding of the definitions?