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Sometimes it can be thought of as the shortcut you mentioned, or as a replacement of 'veamos', but as a set expression it has other meanings (see «a ver» and «a ver si»). Look here (part a), only).

It can be used when you're asking another to show you something.
Use it to express interest in learning something new.
As an exclamation, it means "of course."

If it begins a sentence, followed by the conjunction 'si', it can be used to express a fear or suspicion, an expectation, or a wish/command.

A ver