Is the expansion of the universe proof of the big bang?

In and of itself, no the expansion of the universe is not proof of the big bang. Other theories could be constructed that would also be explained by the expansion. For example Fred Hoyle and others proposed the Steady State Theory which proposed that new matter was constantly being created thus causing the expansion. A the time there was no other evidence for either theory so either could have been true.

However, when you couple the expansion with other evidence such as the cosmic microwave background radiation discovered in 1964 then it does strengthen the case for the big bang being the cause.