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Definition df-cllaw 41610
Description: The closure law for binary operations, see definitions of laws A0. and M0. in section 1.1 of [Hall] p. 1, or definition 1 in [BourbakiAlg1] p. 1: the value of a binary operation applied to two operands of a given sets is an element of this set. By this definition, the closure law is expressed as binary relation: a binary operation is related to a set by clLaw if the closure law holds for this binary operation regarding this set. Note that the binary operation needs not to be a function. (Contributed by AV, 7-Jan-2020.)
Assertion
Ref Expression
df-cllaw clLaw = {⟨𝑜, 𝑚⟩ ∣ ∀𝑥𝑚𝑦𝑚 (𝑥𝑜𝑦) ∈ 𝑚}
Distinct variable group:   𝑚,𝑜,𝑥,𝑦

Detailed syntax breakdown of Definition df-cllaw
StepHypRef Expression
1 ccllaw 41607 . 2 class clLaw
2 vx . . . . . . . 8 setvar 𝑥
32cv 1473 . . . . . . 7 class 𝑥
4 vy . . . . . . . 8 setvar 𝑦
54cv 1473 . . . . . . 7 class 𝑦
6 vo . . . . . . . 8 setvar 𝑜
76cv 1473 . . . . . . 7 class 𝑜
83, 5, 7co 6523 . . . . . 6 class (𝑥𝑜𝑦)
9 vm . . . . . . 7 setvar 𝑚
109cv 1473 . . . . . 6 class 𝑚
118, 10wcel 1975 . . . . 5 wff (𝑥𝑜𝑦) ∈ 𝑚
1211, 4, 10wral 2891 . . . 4 wff 𝑦𝑚 (𝑥𝑜𝑦) ∈ 𝑚
1312, 2, 10wral 2891 . . 3 wff 𝑥𝑚𝑦𝑚 (𝑥𝑜𝑦) ∈ 𝑚
1413, 6, 9copab 4632 . 2 class {⟨𝑜, 𝑚⟩ ∣ ∀𝑥𝑚𝑦𝑚 (𝑥𝑜𝑦) ∈ 𝑚}
151, 14wceq 1474 1 wff clLaw = {⟨𝑜, 𝑚⟩ ∣ ∀𝑥𝑚𝑦𝑚 (𝑥𝑜𝑦) ∈ 𝑚}
Colors of variables: wff setvar class
This definition is referenced by:  iscllaw  41613  clcllaw  41615
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