Definition df-xp 4681

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 4673	. 2 class ( A X. B )
4		vx	. . . . . 6 setvar x
5	4	cv 1372	. . . . 5 class x
6	5, 1	wcel 2176	. . . 4 wff x e. A
7		vy	. . . . . 6 setvar y
8	7	cv 1372	. . . . 5 class y
9	8, 2	wcel 2176	. . . 4 wff y e. B
10	6, 9	wa 104	. . 3 wff ( x e. A /\ y e. B )
11	10, 4, 7	copab 4104	. 2 class { <. x , y >. | ( x e. A /\ y e. B ) }
12	3, 11	wceq 1373	1 wff ( A X. B ) = { <. x , y >. | ( x e. A /\ y e. B ) }

This definition is referenced by:  xpeq1  4689  xpeq2  4690  elxpi  4691  elxp  4692  nfxp  4702  fconstmpt  4722  brab2a  4728  xpundi  4731  xpundir  4732  opabssxp  4749  csbxpg  4756  xpss12  4782  relopabiv  4801  inxp  4812  dmxpm  4898  dmxpid  4899  resopab  5003  cnvxp  5101  xpcom  5229  dfxp3  6280  dmaddpq  7492  dmmulpq  7493  enq0enq  7544  npsspw  7584  shftfvalg  11129  shftfval  11132  eqgfval  13558  reldvdsrsrg  13854  dvdsrvald  13855  dvdsrex  13860  lgsquadlem3  15556