classificatory clauses with bí and der preposition i (= in)

The use of the preposition i = in is easily explained. Because it is a condition that is being described, in which one finds oneself or a function that one has with "in" (= in oneself).

Here, there are 3 clause forms:

1. i + possessive pronoun + predicate:

Tá mé i mo dhochtúir. = I am a doctor. ("I am in my (status as) a doctor")
Tá Bríd ina dochtúir. = Bríd is a doctor. ("Bríd is in-her (function as) a doctor")

2. within a relative clause (i + subject):

Dochtúir atá ionam. = I am a doctor. ("a doctor, that is in-me", meaning: "...whose function is in me")
Dochtúir atá ann. = He is a doctor. ("a doctor, that is in-him", "...whose function is in him")
Dochtúir atá in Bríd. = Bríd is a doctor ("a doctor, that is in Bríd", "... whose function is in Bríd")

Note the use of in (instead of i) here preceding names and the missing eclipsis. (in Bríd instead of i mBríd).
In some grammar explanations, the use of the prepositional pronoun (e.g. ionam) is seen as allowed.
Furthermore, one should note the double meaning of "Dochtúir atá ann": "He is a doctor" and "There is a doctor " (due to the common adverbial use of ann = da: tá .. ann = "there is")

3. in an "only"-clause (i + subject):

Níl ionam ach amadán = I am only an idiot (lit.: not-is in-me but an idiot)

Only an "only"-clause is considerable as a non-relative clause form.
A normal non-relative clause with i + subject, like e.g.: *Tá amadán ionam is not possible!
Here, one could only do so as the form Tá mé i m'amadán = I am an idiot (see 1.)

use:
These forms are used,

• if a recently attained or temporary function, profession or similar is meant.
  e.g.: Bhí mé fós i mo chailín beag ag an am sin = I was still a little girl at that time
  If the connection of subject and predicate is already inborn, and this (permanent) connection cannot be reversed, one cannot use these expressions, then it is only possible as a copular clause:
  (incorrect e.g. *Tá an bradán ina iasc instead of correct: Is iasc é an bradán = The salmon is a fish, because the salmon was always a fish and will always remain one, it's not just temporarily taking on the job of a fish)
• in tenses where there is no copular form, so, in the future, imperative, as well as in clauses requiring a verbal noun. Most of the time these sentences are logically equivalent to the first point. e.g.:
  Bí i d'fhear! = Be a man! (lit.: "be in your man"),
  Ní maith léi bheith ina cailín beag. = She doesn't like being a little girl. (lit.: "not-is good with-her being in-her girl small ")
  

classificatory clause with bí and the preposition ar (= on)

This expression is used especially in those cases when "one of...", and the like should be expressed with the superlative. It is also used for statements like "one so ... as ... " (i.e. as the attributive equative). e.g.:

Tá sé ar na daoine is mó sa rang = He is one of the biggest in the class.(lit.: "is he on the people are biggest in the class ")
Tá sí ar na cailíní is áille ar an domhan = She is one of the most beautiful girls in the world. (lit.: "is she on the girls are most beautiful on the world ")
Tá sé ar fhear chomh maith leat = He is a man as good as you.
Tá sin ar ceann de na rudaí a dúirt mé = That is one of the things of which I spoke.(lit.: "is that on head of the things that spoke I")

identificatory clauses with bí and the preposition ar (= on)

These are mostly used to demonstrate the superlative ("the smartest " etc.) and with ordinal numbers ("the first ", "the second ", etc.).

Tá sé ar an bhfear is cliste sa bhaile = He is the smartest man in the town. (lit.: "is he on the man is smartest in-the town ")
Tá sí ar an gcailín is áille in Éirinn = She is the most beautiful girl in Ireland (lit.: "is she on the girl is most beautiful in Ireland")
Bhí sé ar an dara fear a tháinig = He was the second man that came (lit.: "was he on the second man that came ")