If we had a global collapse, as you postulate, there would be a period of riots and hoarding, followed by foraging and scavenging the last remaining, working pieces of technology or safety. There would be many, many deaths, either from violence or through inability to survive, but there are more than enough people on the planet who can (and do) survive with little to no technological help, and would be able to pass on their genes and teach their children what they need to know.
Eventually, a new society would form, and depending on how severe the original collapse was, how long the recovery took, and where the recovery took place, the "new" society could either be pretty similar to what we have now, or it could be entirely different. As a species, we are very adaptable, very intelligent, and very good at solving complex problems.
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