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Definition df-coss 38413
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 38415 and the comment of dfec2 8749). 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 38853). Without the definition of 𝑅 we should have to relate the right side of these theorems to a composition of a converse (cf. dfcoss3 38416) or to the range of a range Cartesian product of classes (cf. dfcoss4 38417), which would make the theorems complicated and confusing. Alternate definition is dfcoss2 38415. Technically, we can define it via composition (dfcoss3 38416) or as the range of a range Cartesian product (dfcoss4 38417), but neither of these definitions reveal directly how the cosets by 𝑅 relate to each other. We define functions (df-funsALTV 38683, df-funALTV 38684) and disjoints (dfdisjs 38710, dfdisjs2 38711, df-disjALTV 38707, dfdisjALTV2 38716) 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 38183 . 2 class 𝑅
3 vu . . . . . . 7 setvar 𝑢
43cv 1538 . . . . . 6 class 𝑢
5 vx . . . . . . 7 setvar 𝑥
65cv 1538 . . . . . 6 class 𝑥
74, 6, 1wbr 5142 . . . . 5 wff 𝑢𝑅𝑥
8 vy . . . . . . 7 setvar 𝑦
98cv 1538 . . . . . 6 class 𝑦
104, 9, 1wbr 5142 . . . . 5 wff 𝑢𝑅𝑦
117, 10wa 395 . . . 4 wff (𝑢𝑅𝑥𝑢𝑅𝑦)
1211, 3wex 1778 . . 3 wff 𝑢(𝑢𝑅𝑥𝑢𝑅𝑦)
1312, 5, 8copab 5204 . 2 class {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝑅𝑥𝑢𝑅𝑦)}
142, 13wceq 1539 1 wff 𝑅 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑢(𝑢𝑅𝑥𝑢𝑅𝑦)}
Colors of variables: wff setvar class
This definition is referenced by:  dfcoss2  38415  dfcoss3  38416  dfcoss4  38417  cosscnv  38418  coss1cnvres  38419  relcoss  38425  cossss  38427  cosseq  38428  1cossres  38431  brcoss  38433  cossssid2  38470  cossid  38482
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