Definition df-xp 5691

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⟩}) (ex-xp 30455). Another example is that the set of rational numbers is defined in df-q 12991 using the Cartesian product (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 5683	. 2 class (𝐴 × 𝐵)
4		vx	. . . . . 6 setvar 𝑥
5	4	cv 1539	. . . . 5 class 𝑥
6	5, 1	wcel 2108	. . . 4 wff 𝑥 ∈ 𝐴
7		vy	. . . . . 6 setvar 𝑦
8	7	cv 1539	. . . . 5 class 𝑦
9	8, 2	wcel 2108	. . . 4 wff 𝑦 ∈ 𝐵
10	6, 9	wa 395	. . 3 wff (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)
11	10, 4, 7	copab 5205	. 2 class {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)}
12	3, 11	wceq 1540	1 wff (𝐴 × 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)}

This definition is referenced by:  xpeq1  5699  xpss12  5700  xpeq2  5706  elxpi  5707  elxp  5708  nfxp  5718  fconstmpt  5747  xpundi  5754  xpundir  5755  elopaelxp  5775  opabssxp  5778  csbxp  5785  ssrel  5792  ssrelOLD  5793  relopabiv  5830  relopabi  5832  inxpOLD  5843  dmxpOLD  5940  resopab  6052  cnvxp  6177  xpco  6309  1st2val  8042  2nd2val  8043  dfxp3  8086  marypha2lem2  9476  wemapwe  9737  cardf2  9983  dfac3  10161  axdc2lem  10488  fpwwe2lem1  10671  canthwe  10691  xpcogend  15013  shftfval  15109  ipoval  18575  ipolerval  18577  eqgfval  19194  frgpuplem  19790  pjfval2  21729  ltbwe  22062  opsrtoslem1  22079  2ndcctbss  23463  ulmval  26423  lgsquadlem3  27426  iscgrg  28520  ishpg  28767  nvss  30612  ajfval  30828  fpwrelmap  32744  afsval  34686  cvmlift2lem12  35319  bj-opabssvv  37151  bj-xpcossxp  37190  dicval  41178  dnwech  43060  fgraphopab  43215  areaquad  43228  csbxpgVD  44914  relopabVD  44921  dfnelbr2  47285  xpsnopab  48073