رقم الرف في فهرس المكتبة.

It is built from:

• رقم الرف = an iḍāfa (possessive/genitive construction): the shelf number / literally the number of the shelf
• في = in
• فهرس المكتبة = another iḍāfa: the library catalog / literally the catalog of the library

So the whole expression is basically:

• [رقم الرف] + [في فهرس المكتبة]

Depending on context, it can be read as:

• a noun phrase / label on a form or in a system, or
• a nominal sentence with an understood present-tense is

Arabic often allows both kinds of reading when there is no written vowel marking and little context.

Because Arabic often leaves out the verb to be in the present tense.

So if this is being read as a full sentence, the meaning is understood as something like:

• The shelf number is in the library catalog

There is no separate word for is in this kind of present-tense sentence.

Also, in real life, expressions like this are very often just labels, headings, or database-style phrases, where no verb is needed at all.

Because فهرس المكتبة is an iḍāfa.

In an iḍāfa:

• the first noun usually does not take ال
• the second noun determines whether the whole phrase is definite

So:

• فهرس المكتبة = the catalog of the library = the library catalog

Even though فهرس has no ال, the whole phrase is definite because المكتبة is definite.

The same pattern appears in:

• رقم الرف
  Here رقم has no ال, but the whole phrase is definite because الرف is definite.

That is the Arabic iḍāfa construction, which often corresponds to English of or to a noun-noun phrase.

So:

• رقم الرف literally = number of the shelf
• natural English = the shelf number

Arabic uses this construction constantly. Some similar examples:

• باب البيت = the door of the house / the house door
• مدير المدرسة = the principal of the school / the school principal

So رقم الرف is completely normal Arabic word order.

Here is the definiteness pattern:

• الرف is definite because it has ال
• المكتبة is definite because it has ال
• رقم and فهرس do not show ال, but each is the first part of an iḍāfa

In Arabic, the first noun in an iḍāfa takes its definiteness from the second noun.

So:

• رقم الرف is definite
• فهرس المكتبة is definite

That is why the whole expression refers to specific things, not just any shelf or any library.

في is the preposition in.

It connects رقم الرف with فهرس المكتبة and shows location or inclusion, depending on context.

A very important grammar point:

• a noun after في is in the genitive case

So فهرس is grammatically genitive here.

If full case endings were written, you could see that more clearly.

A common fully vowelled version would be:

• رَقْمُ الرَّفِّ في فِهْرِسِ المَكْتَبَةِ

If you want to see the grammar behind that:

• رَقْمُ: first word, often nominative if the phrase is standing on its own as the topic
• الرَّفِّ: genitive because it is the second noun in an iḍāfa
• فِهْرِسِ: genitive because it comes after في
• المَكْتَبَةِ: genitive because it is the second noun in an iḍāfa

One useful note: in normal modern writing, these short vowels are usually omitted, so learners have to infer them from grammar.

Because nouns in an iḍāfa do not take tanwīn on the first noun.

So:

• رقم in رقم الرف has no tanwīn because it is the first part of an iḍāfa
• فهرس in فهرس المكتبة also has no tanwīn because it is the first part of an iḍāfa

This is one of the signs that you are looking at an iḍāfa structure.

In الرف, the ل of ال is not pronounced normally, because ر is a sun letter.

So الرف is pronounced roughly:

• ar-raff

not:

• al-raff

That doubling sound on r is important.

By contrast, in المكتبة, the ل is pronounced normally because م is a moon letter:

• al-maktaba

So a simple pronunciation guide for the whole phrase is:

• raqm ar-raff fī fihris al-maktaba

In pause, final case vowels are often not pronounced.

Both are valid ways to understand it.

• Literal structure: number of the shelf
• Natural English rendering: shelf number

This is very common when translating Arabic iḍāfa constructions into English: the most literal version and the most natural version are not always identical, but they refer to the same structure.

Not necessarily.

Without more context, it could be:

• a complete nominal sentence with an understood is
• a heading
• a form label
• a catalog/database phrase

Arabic often uses short nominal expressions like this where English might prefer a full sentence.

So the exact analysis depends on where you saw it. Grammatically, though, the internal structure remains the same.