CriteriaBuilder, specification and subquery

Question

I use spring boot 3

School have a one to many relation with Teacher

I created 2 specfication

public Page<School> advanceSearch(SchoolSearch search, Pageable page) {


    Specification<School> hasCityName = (Root<School> mainRoot, CriteriaQuery<?> mainCq, CriteriaBuilder mainCb) -> {
    ...
        Predicate predicate..
        return predicate;
    };

    Specification<Teacher> hasTeacherName = (Root<Teacher> mainRoot, CriteriaQuery<?> mainCq, CriteriaBuilder mainCb) -> {
    ...
        Predicate predicate..
        return predicate;
    };

    return findAll(hasCityName, page);
}

I search to do

select s  from School s where s.city like city and 
s.schoolId in
(select t.school.schooId from Teacher t where t.name like teacherNom)

how to do a in condition with specfication?

I tried with no success

     Specification<Teacher> fromRepondant = (Root<Teacher> root, CriteriaQuery<?> cq, CriteriaBuilder cb) -> {
        
        Predicate pre = cb.and(cb.equal(root.get("name"), search.name()));


       Path<School> schooPath = root.get("school");

       return cq.select(schooPath.get("schoolId")).from(School.class).in(pre);
        
    };

     Specification<School> hasIdIn = (Root<School> root, CriteriaQuery<?> cq, CriteriaBuilder cb) -> {
    
        return cb.and(root.get("schooId").in(hasTeacherName));

    };