Definition df-xp 4724

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 >. } ). Another example is that the set of rational numbers is defined using the Cartesian product as ( 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.)

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 4716	. 2 class ( A X. B )
4		vx	. . . . . 6 setvar x
5	4	cv 1394	. . . . 5 class x
6	5, 1	wcel 2200	. . . 4 wff x e. A
7		vy	. . . . . 6 setvar y
8	7	cv 1394	. . . . 5 class y
9	8, 2	wcel 2200	. . . 4 wff y e. B
10	6, 9	wa 104	. . 3 wff ( x e. A /\ y e. B )
11	10, 4, 7	copab 4143	. 2 class { <. x , y >. | ( x e. A /\ y e. B ) }
12	3, 11	wceq 1395	1 wff ( A X. B ) = { <. x , y >. | ( x e. A /\ y e. B ) }

This definition is referenced by:  xpeq1  4732  xpeq2  4733  elxpi  4734  elxp  4735  nfxp  4745  fconstmpt  4765  brab2a  4771  xpundi  4774  xpundir  4775  opabssxp  4792  csbxpg  4799  xpss12  4825  relopabiv  4844  inxp  4855  dmxpm  4943  dmxpid  4944  resopab  5048  cnvxp  5146  xpcom  5274  dfxp3  6338  dmaddpq  7562  dmmulpq  7563  enq0enq  7614  npsspw  7654  shftfvalg  11324  shftfval  11327  eqgfval  13754  reldvdsrsrg  14050  dvdsrvald  14051  dvdsrex  14056  lgsquadlem3  15752