Definition df-xp 4419

Description: Define the cross product of two classes. Definition 9.11 of [Quine] p. 64. For example, ( { 1 , 5 } X. { 2 , 7 } ) = ( { <. 1 , 2 >., <. 1 , 7 >. } u. { <. 5 , 2 >., <. 5 , 7 >. } ) . Another example is that the set of rational numbers are defined in using the cross-product ( Z X. N ) ; the left- and right-hand sides of the cross-product represent the top (integer) and bottom (natural) numbers of a fraction. (Contributed by NM, 4-Jul-1994.)

Assertion

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

Distinct variable groups:   x, y, A   x, B, y

Detailed syntax breakdown of Definition df-xp

Step	Hyp	Ref	Expression
1		cA	. . 3 class A
2		cB	. . 3 class B
3	1, 2	cxp 4411	. 2 class ( A X. B )
4		vx	. . . . . 6 setvar x
5	4	cv 1286	. . . . 5 class x
6	5, 1	wcel 1436	. . . 4 wff x e. A
7		vy	. . . . . 6 setvar y
8	7	cv 1286	. . . . 5 class y
9	8, 2	wcel 1436	. . . 4 wff y e. B
10	6, 9	wa 102	. . 3 wff ( x e. A /\ y e. B )
11	10, 4, 7	copab 3875	. 2 class { <. x , y >. | ( x e. A /\ y e. B ) }
12	3, 11	wceq 1287	1 wff ( A X. B ) = { <. x , y >. | ( x e. A /\ y e. B ) }

This definition is referenced by:  xpeq1  4427  xpeq2  4428  elxpi  4429  elxp  4430  nfxp  4439  fconstmpt  4455  brab2a  4461  xpundi  4464  xpundir  4465  opabssxp  4482  csbxpg  4489  xpss12  4515  inxp  4540  dmxpm  4626  resopab  4725  cnvxp  4818  xpcom  4945  dfxp3  5923  dmaddpq  6885  dmmulpq  6886  enq0enq  6937  npsspw  6977  shftfvalg  10152  shftfval  10155