|Description: Define the class of
cosets by 𝑅: 𝑥 and 𝑦 are cosets by
𝑅 iff there exists a set 𝑢 such
that both 𝑢𝑅𝑥 and
𝑢𝑅𝑦 hold, i.e., both 𝑥 and
are are elements of the 𝑅
-coset of 𝑢 (cf. dfcoss2 34793 and the comment of dfec2 8029). 𝑅 is
usually a relation.
This concept simplifies theorems relating partition and equivalence: the
left side of these theorems relate to 𝑅, the right side relate to
≀ 𝑅 (cf. e.g. ~? pet ). Without the
definition of ≀ 𝑅 we
should have to relate the right side of these theorems to a composition
of a converse (cf. dfcoss3 34794) or to the range of a range Cartesian
product of classes (cf. dfcoss4 34795), which would make the theorems
complicated and confusing. Alternate definition is dfcoss2 34793.
Technically, we can define it via composition (dfcoss3 34794) or as the
range of a range Cartesian product (dfcoss4 34795), but neither of these
definitions reveal directly how the cosets by 𝑅 relate to each
other. We define functions ( ~? df-funsALTV , ~? df-funALTV ) and
disjoints ( ~? dfdisjs , ~? dfdisjs2 , ~? df-disjALTV , ~?
dfdisjALTV2 ) with the help of it as well. (Contributed by Peter Mazsa,