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Definition df-coss 34504
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 (cf. dfcoss2 34506 and the comment of dfec2 7898). 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 34507) or to the range of a range Cartesian product of classes (cf. dfcoss4 34508), which would make the theorems complicated and confusing. Alternate definition is dfcoss2 34506. Technically, we can define it via composition (dfcoss3 34507) or as the range of a range Cartesian product (dfcoss4 34508), 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, 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 34308 . 2 class 𝑅
3 vu . . . . . . 7 setvar 𝑢
43cv 1629 . . . . . 6 class 𝑢
5 vx . . . . . . 7 setvar 𝑥
65cv 1629 . . . . . 6 class 𝑥
74, 6, 1wbr 4784 . . . . 5 wff 𝑢𝑅𝑥
8 vy . . . . . . 7 setvar 𝑦
98cv 1629 . . . . . 6 class 𝑦
104, 9, 1wbr 4784 . . . . 5 wff 𝑢𝑅𝑦
117, 10wa 382 . . . 4 wff (𝑢𝑅𝑥𝑢𝑅𝑦)
1211, 3wex 1851 . . 3 wff 𝑢(𝑢𝑅𝑥𝑢𝑅𝑦)
1312, 5, 8copab 4844 . 2 class {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝑅𝑥𝑢𝑅𝑦)}
142, 13wceq 1630 1 wff 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝑅𝑥𝑢𝑅𝑦)}
Colors of variables: wff setvar class
This definition is referenced by:  dfcoss2  34506  dfcoss3  34507  dfcoss4  34508  relcoss  34513  cossss  34515  cosseq  34516  1cossres  34519  brcoss  34521  cossssid2  34553  cossid  34565
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