Definition df-xp 4670

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} × {2, 7}) = ({⟨1, 2⟩, ⟨1, 7⟩} ∪ {⟨5, 2⟩, ⟨5, 7⟩}). Another example is that the set of rational numbers is defined using the Cartesian product as (ℤ × ℕ); 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 × 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 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 4662	. 2 class (𝐴 × 𝐵)
4		vx	. . . . . 6 setvar 𝑥
5	4	cv 1363	. . . . 5 class 𝑥
6	5, 1	wcel 2167	. . . 4 wff 𝑥 ∈ 𝐴
7		vy	. . . . . 6 setvar 𝑦
8	7	cv 1363	. . . . 5 class 𝑦
9	8, 2	wcel 2167	. . . 4 wff 𝑦 ∈ 𝐵
10	6, 9	wa 104	. . 3 wff (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)
11	10, 4, 7	copab 4094	. 2 class {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)}
12	3, 11	wceq 1364	1 wff (𝐴 × 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)}

This definition is referenced by:  xpeq1  4678  xpeq2  4679  elxpi  4680  elxp  4681  nfxp  4691  fconstmpt  4711  brab2a  4717  xpundi  4720  xpundir  4721  opabssxp  4738  csbxpg  4745  xpss12  4771  relopabiv  4790  inxp  4801  dmxpm  4887  dmxpid  4888  resopab  4991  cnvxp  5089  xpcom  5217  dfxp3  6261  dmaddpq  7465  dmmulpq  7466  enq0enq  7517  npsspw  7557  shftfvalg  11002  shftfval  11005  eqgfval  13430  reldvdsrsrg  13726  dvdsrvald  13727  dvdsrex  13732  lgsquadlem3  15428