Definition df-xp 4725

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 4717	. 2 class (𝐴 × 𝐵)
4		vx	. . . . . 6 setvar 𝑥
5	4	cv 1394	. . . . 5 class 𝑥
6	5, 1	wcel 2200	. . . 4 wff 𝑥 ∈ 𝐴
7		vy	. . . . . 6 setvar 𝑦
8	7	cv 1394	. . . . 5 class 𝑦
9	8, 2	wcel 2200	. . . 4 wff 𝑦 ∈ 𝐵
10	6, 9	wa 104	. . 3 wff (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)
11	10, 4, 7	copab 4144	. 2 class {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)}
12	3, 11	wceq 1395	1 wff (𝐴 × 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)}

This definition is referenced by:  xpeq1  4733  xpeq2  4734  elxpi  4735  elxp  4736  nfxp  4746  fconstmpt  4766  brab2a  4772  xpundi  4775  xpundir  4776  opabssxp  4793  csbxpg  4800  xpss12  4826  relopabiv  4845  inxp  4856  dmxpm  4944  dmxpid  4945  resopab  5049  cnvxp  5147  xpcom  5275  dfxp3  6346  dmaddpq  7574  dmmulpq  7575  enq0enq  7626  npsspw  7666  shftfvalg  11337  shftfval  11340  eqgfval  13767  dvdsrvald  14065  dvdsrex  14070  lgsquadlem3  15766