MongoDB match array based on document value

Question

Let's say I have the following document structure:

{
  "A": {
    "_id": "ID0"
  },
  "B": [
    "a": {
      "_id": "ID0",
      "field": "X"
    },
    "b": {
      "_id": "ID1",
      "field": "Y"
    }
  ]
}

I want to project B matched with the _id in A. The end result would be:

{
  "B": [
    "a": {
      "_id": "ID0",
      "field": "X"
    }
  ]
}

I tried the following but apparently I'm doing something wrong:

db.collection.aggregate([
  {$match: {"B._id": "$A._id"}},
  {$project: {"B": 1}}
])