Meaning of "auf" in the following context

Question

Came across this sentence in a Mozart libretto and was curious what the meaning of "auf" is here.

Auf den Rat, den ich gegeben,
Sei, mein Kind, mit Fleiss bedacht.

Google Translate translates this as:

On the advice I gave
Be diligent, my child.

So seems the meaning of "auf" here is "regarding" or "about". However, I've never come across this usage of "auf" in German before. Is this an archaic use of the preposition?

Answer 1

Here, the preposition auf belongs to the fixed phrase

• auf etwas bedacht sein

You can see this if you simplify and rearrange the verse:

Sei auf den Rat bedacht.
_{(Be mindful of/attentive to the advice.)}

A close translation of the verse could therefore be as follows:

Of the advice I have given,
be mindful with diligence, my child.

Answer 2

No, it's a causal adverbial that is in common use.

Auf deinen Wunsch (hin) habe ich das noch heute erledigt.

It's often a part of a phrasal verb as for example antworten auf:

Auf den Brief habe ich aber noch nicht geantwortet.

Or warten auf:

Ich warte auf den Bus.