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 >. | A. x e. m A. y e. m ( x o y ) e. 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 ) e. m
11	10 5 4	wral	|- A. y e. m ( x o y ) e. m
12	11 3 4	wral	|- A. x e. m A. y e. m ( x o y ) e. m
13	12 1 2	copab	|- { <. o , m >. | A. x e. m A. y e. m ( x o y ) e. m }
14	0 13	wceq	|- clLaw = { <. o , m >. | A. x e. m A. y e. m ( x o y ) e. m }