Definition df-xp

Description: Define the Cartesian product of two classes. This is also sometimes called the "cross product" but that term also has other meanings; we intentionally choose a less ambiguous term. Definition 9.11 of Quine p. 64. For example, ( { 1 , 5 } X. { 2 , 7 } ) = ( { <. 1 , 2 >. , <. 1 , 7 >. } u. { <. 5 , 2 >. , <. 5 , 7 >. } ) ( ex-xp ). Another example is that the set of rational numbers is defined in df-q using the Cartesian product ( ZZ X. NN ) ; the left- and right-hand sides of the Cartesian product represent the top (integer) and bottom (natural) numbers of a fraction. (Contributed by NM, 4-Jul-1994)

	Ref	Expression
Assertion	df-xp	|- ( A X. B ) = { <. x , y >. | ( x e. A /\ y e. B ) }

Detailed syntax breakdown

Step	Hyp	Ref	Expression
0		cA	|- A
1		cB	|- B
2	0 1	cxp	|- ( A X. B )
3		vx	|- x
4		vy	|- y
5	3	cv	|- x
6	5 0	wcel	|- x e. A
7	4	cv	|- y
8	7 1	wcel	|- y e. B
9	6 8	wa	|- ( x e. A /\ y e. B )
10	9 3 4	copab	|- { <. x , y >. | ( x e. A /\ y e. B ) }
11	2 10	wceq	|- ( A X. B ) = { <. x , y >. | ( x e. A /\ y e. B ) }