Definition df-cllaw

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)

	Ref	Expression
Assertion	df-cllaw	⊢ clLaw = { ⟨ 𝑜 , 𝑚 ⟩ ∣ ∀ 𝑥 ∈ 𝑚 ∀ 𝑦 ∈ 𝑚 ( 𝑥 𝑜 𝑦 ) ∈ 𝑚 }

Detailed syntax breakdown

Step	Hyp	Ref	Expression
0		ccllaw	⊢ clLaw
1		vo	⊢ 𝑜
2		vm	⊢ 𝑚
3		vx	⊢ 𝑥
4	2	cv	⊢ 𝑚
5		vy	⊢ 𝑦
6	3	cv	⊢ 𝑥
7	1	cv	⊢ 𝑜
8	5	cv	⊢ 𝑦
9	6 8 7	co	⊢ ( 𝑥 𝑜 𝑦 )
10	9 4	wcel	⊢ ( 𝑥 𝑜 𝑦 ) ∈ 𝑚
11	10 5 4	wral	⊢ ∀ 𝑦 ∈ 𝑚 ( 𝑥 𝑜 𝑦 ) ∈ 𝑚
12	11 3 4	wral	⊢ ∀ 𝑥 ∈ 𝑚 ∀ 𝑦 ∈ 𝑚 ( 𝑥 𝑜 𝑦 ) ∈ 𝑚
13	12 1 2	copab	⊢ { ⟨ 𝑜 , 𝑚 ⟩ ∣ ∀ 𝑥 ∈ 𝑚 ∀ 𝑦 ∈ 𝑚 ( 𝑥 𝑜 𝑦 ) ∈ 𝑚 }
14	0 13	wceq	⊢ clLaw = { ⟨ 𝑜 , 𝑚 ⟩ ∣ ∀ 𝑥 ∈ 𝑚 ∀ 𝑦 ∈ 𝑚 ( 𝑥 𝑜 𝑦 ) ∈ 𝑚 }