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Definition df-coss 38057
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 𝑦 are are elements of the 𝑅 -coset of 𝑢 (see dfcoss2 38059 and the comment of dfec2 8736). 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 (see e.g. pet 38497). Without the definition of 𝑅 we should have to relate the right side of these theorems to a composition of a converse (cf. dfcoss3 38060) or to the range of a range Cartesian product of classes (cf. dfcoss4 38061), which would make the theorems complicated and confusing. Alternate definition is dfcoss2 38059. Technically, we can define it via composition (dfcoss3 38060) or as the range of a range Cartesian product (dfcoss4 38061), but neither of these definitions reveal directly how the cosets by 𝑅 relate to each other. We define functions (df-funsALTV 38327, df-funALTV 38328) and disjoints (dfdisjs 38354, dfdisjs2 38355, df-disjALTV 38351, dfdisjALTV2 38360) with the help of it as well. (Contributed by Peter Mazsa, 9-Jan-2018.)

Assertion
Ref Expression
df-coss 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝑅𝑥𝑢𝑅𝑦)}
Distinct variable group:   𝑢,𝑅,𝑥,𝑦

Detailed syntax breakdown of Definition df-coss
StepHypRef Expression
1 cR . . 3 class 𝑅
21ccoss 37824 . 2 class 𝑅
3 vu . . . . . . 7 setvar 𝑢
43cv 1532 . . . . . 6 class 𝑢
5 vx . . . . . . 7 setvar 𝑥
65cv 1532 . . . . . 6 class 𝑥
74, 6, 1wbr 5152 . . . . 5 wff 𝑢𝑅𝑥
8 vy . . . . . . 7 setvar 𝑦
98cv 1532 . . . . . 6 class 𝑦
104, 9, 1wbr 5152 . . . . 5 wff 𝑢𝑅𝑦
117, 10wa 394 . . . 4 wff (𝑢𝑅𝑥𝑢𝑅𝑦)
1211, 3wex 1773 . . 3 wff 𝑢(𝑢𝑅𝑥𝑢𝑅𝑦)
1312, 5, 8copab 5214 . 2 class {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝑅𝑥𝑢𝑅𝑦)}
142, 13wceq 1533 1 wff 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝑅𝑥𝑢𝑅𝑦)}
Colors of variables: wff setvar class
This definition is referenced by:  dfcoss2  38059  dfcoss3  38060  dfcoss4  38061  cosscnv  38062  coss1cnvres  38063  relcoss  38069  cossss  38071  cosseq  38072  1cossres  38075  brcoss  38077  cossssid2  38114  cossid  38126
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