bi on dwieneun gyedani mikkeureowoseo josimhaeya haeyo.

온 is the past adnominal form of 오다 in 비가 오다 (to rain).

So:

• 비 온 뒤 = after it rained / after the rain
• 비 오는 would mean raining or that is raining, which does not fit naturally with 뒤 here

Korean often uses the past adnominal form before 뒤 to mean after doing something:

• 밥 먹은 뒤 = after eating
• 수업 끝난 뒤 = after class ended
• 비 온 뒤 = after it rained

A fuller version would be 비가 온 뒤.

Yes, 비가 온 뒤 is the full and very standard form.

In everyday Korean, particles like 이/가 are often omitted when the meaning is obvious. So:

• 비가 온 뒤 = full form
• 비 온 뒤 = natural shortened form

Both are understood as after it rains / after the rain.

This kind of omission is very common in speech and informal writing.

뒤에 means after, later than, or after the time when.

So:

• 비 온 뒤에 = after it rains

When 는 is added:

• 뒤에는

it gives a topic or slight contrast feeling, like:

• as for after it rains...
• after it rains, ...

In this sentence, 뒤에는 helps set the situation or condition for the rest of the sentence.

Compare:

• 비 온 뒤에 계단이 미끄러워서... = after it rains, the stairs are slippery...
• 비 온 뒤에는 계단이 미끄러워서... = as for after it rains, the stairs are slippery...

The second one sounds a little more natural as a general statement.

Here, 계단이 marks 계단 (stairs) as the subject of 미끄럽다 (to be slippery).

So the structure is basically:

• 계단이 미끄러워서 = the stairs are slippery, so...

Why not the others?

• 계단이 = the stairs are the thing that is slippery
• 계단은 = would add contrast, like as for the stairs, they are slippery
• 계단을 = object marker, which would be wrong here because 미끄럽다 does not take a direct object in this meaning

So 계단이 is the neutral and correct choice.

미끄러워서 comes from 미끄럽다 (to be slippery).

The ending -아/어서 often means:

• because
• so
• sometimes simple connection between clauses

Here it expresses a reason/cause:

• 계단이 미끄러워서 조심해야 해요
• The stairs are slippery, so you have to be careful

So 미끄러워서 means because it’s slippery or it’s slippery, so...

It comes from:

• 조심하다 = to be careful / to take care
• -아/어야 하다 = must / have to / should

So:

• 조심해야 하다 = must be careful / have to be careful
• 조심해야 해요 = polite spoken form

This pattern is very common:

• 가야 해요 = have to go
• 먹어야 해요 = have to eat
• 공부해야 해요 = have to study

In everyday English, -아/어야 해요 can sound like have to, need to, or should, depending on context.

It is often a bit softer than English must.

Depending on context, 조심해야 해요 can mean:

• you have to be careful
• you should be careful
• you need to be careful

In this sentence, it sounds like practical advice or a general warning, not a harsh command.

If someone wanted to sound more direct, they might say:

• 조심하세요 = Be careful.
• 꼭 조심해야 해요 = You really have to be careful.

The sentence ends in 해요, so it is in the polite informal style, often called the -요 style.

That means it is appropriate in many everyday situations.

Compare:

• 조심해야 해요 = polite everyday speech
• 조심해야 합니다 = more formal
• 조심해야 해 = casual, plain speech

So this sentence is polite and natural for normal conversation, announcements, or advice.

Yes. 뒤 and 후 can both mean after.

So these are both possible:

• 비 온 뒤에는
• 비가 온 후에는

The difference is mostly in tone:

• 뒤 sounds more common and conversational
• 후 sounds a bit more formal or written

Also, 비가 온 후에는 sounds slightly fuller and more formal than 비 온 뒤에는.

The sentence can be broken down like this:

• 비 온 뒤에는 = after it rains
• 계단이 = the stairs
• 미끄러워서 = are slippery, so / because they are slippery
• 조심해야 해요 = you have to be careful

So the overall pattern is:

time/situation + reason + advice/necessity

A very literal breakdown would be:

• As for after rain, the stairs are slippery, so one must be careful.

A more natural English version is:

• After it rains, the stairs get slippery, so you have to be careful.