Μια άλλη προϋπόθεση είναι να υπάρχει αποδεικτικό που να δείχνει τη μόνιμη διεύθυνσή σου.

Because προϋπόθεση is a feminine singular noun, and Greek adjectives/articles must agree with the noun.

• μια = a / one (feminine singular)
• άλλη = other / another (feminine singular)
• προϋπόθεση = requirement / condition (feminine singular)

So Μια άλλη προϋπόθεση literally means one other requirement, which is how Greek says another requirement.

Because Modern Greek does not normally use an infinitive the way English does.
Where English says to exist, to show, to go, Greek usually uses να + a finite verb form.

So:

• English: Another requirement is to have / for there to be...
• Greek: Μια άλλη προϋπόθεση είναι να υπάρχει...

Here να υπάρχει functions in the place where English would often use an infinitive.

Not here. In this sentence, είναι is followed by a να-clause that explains what the requirement is.

A helpful way to read it is:

• Η λύση είναι να φύγουμε. = The solution is to leave.
• Μια άλλη προϋπόθεση είναι να υπάρχει... = Another requirement is that there be...

So this is not really the separate idiom είναι να... meaning be supposed to...
It is simply είναι + να-clause as the complement of είναι.

Because υπάρχει here is the imperfective/present subjunctive, which fits the idea of there being / existing / being available.

• να υπάρχει = there should be / there to be / to exist
• να υπάρξει would sound more like to come into existence once or to occur

In this sentence, the document is not being talked about as a one-time event of appearing; it is being described as something that needs to be there. So να υπάρχει is the natural choice.

Αποδεικτικό is a neuter noun meaning something like proof, supporting document, or documentary evidence.

In formal or bureaucratic Greek, it often stands for something like:

• αποδεικτικό έγγραφο = supporting document / proof document

So in this sentence, αποδεικτικό means a document that can serve as proof.

Because it is indefinite here: the sentence means some proof / a supporting document, not the specific document.

Compare:

• να υπάρχει αποδεικτικό = there should be a supporting document / some proof
• να υπάρχει το αποδεικτικό = the supporting document should be there

No article makes it more general and non-specific.

Here που is the relative word meaning that / which.

So:

• αποδεικτικό που... = a document/proof that...

A useful thing to remember is that Greek που does not change form for gender, number, or case the way English who/which/that sometimes does. It is a very common all-purpose relative word.

This is a very common Greek pattern.
After an indefinite or non-specific noun, Greek often uses που να + verb to mean something like:

• that would...
• that should...
• that can...

So:

• αποδεικτικό που να δείχνει τη μόνιμη διεύθυνσή σου = a supporting document that shows / that can show / that would show your permanent address

If you said που δείχνει, it would sound more like you are talking about a document already identified as one that shows the address.
With που να δείχνει, the speaker is describing the kind of document needed.

Because δείχνει here is the imperfective form, which suits a continuing characteristic or function: the document is one that shows your address.

• να δείχνει = to be showing / to show as a characteristic
• να δείξει = to show once, as a complete act

A document’s job is not a single momentary action; it has the property of showing the address. So να δείχνει is the more natural form.

Also, remember that in Modern Greek the present subjunctive often looks exactly like the present indicative; the να is what signals the subjunctive use.

Because the final -ν of την is often dropped before many consonants in Modern Greek.

So both of these may be seen, depending on style and pronunciation habits:

• τη μόνιμη διεύθυνση
• την μόνιμη διεύθυνση

Before μ in μόνιμη, dropping the -ν is very common, so τη μόνιμη looks perfectly normal.

Several things are happening here:

1. Greek usually keeps the definite article with possessives.
   So Greek says:

   • το σπίτι μου = my house
   • η διεύθυνσή σου = your address

   That is why you see τη even though σου already means your.

2. μόνιμη agrees with διεύθυνση.
   διεύθυνση is feminine singular, and here it is the object of δείχνει, so the adjective matches it:

   • τη μόνιμη διεύθυνση = the permanent address

3. σου is the weak possessive form meaning your.
   Grammatically, it is a clitic form historically connected with the genitive.

So the whole phrase means your permanent address, but Greek builds it as the permanent address your.

σου normally comes after the noun in Greek because possessive clitics usually follow the word they belong to:

• το βιβλίο μου = my book
• η διεύθυνσή σου = your address

The extra accent in διεύθυνσή σου is a stress rule. When a word like διεύθυνση is followed by an enclitic word such as σου, Greek often adds an extra written accent to show the pronunciation pattern:

• διεύθυνση
• διεύθυνσή σου

This does not change the meaning or case; it is mainly about correct stress in writing.

The basic meaning would stay the same, but Greek does allow some flexibility in word order.

The original sentence is natural and formal:

• Μια άλλη προϋπόθεση είναι να υπάρχει αποδεικτικό που να δείχνει τη μόνιμη διεύθυνσή σου.

You could rearrange parts for emphasis, but the original order is the most neutral for official language. Greek often keeps this kind of main statement first, details after structure in formal writing.