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Definition df-xp 4629
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⟩ ∣ (𝑥𝐴𝑦𝐵)}
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦

Detailed syntax breakdown of Definition df-xp
StepHypRef Expression
1 cA . . 3 class 𝐴
2 cB . . 3 class 𝐵
31, 2cxp 4621 . 2 class (𝐴 × 𝐵)
4 vx . . . . . 6 setvar 𝑥
54cv 1352 . . . . 5 class 𝑥
65, 1wcel 2148 . . . 4 wff 𝑥𝐴
7 vy . . . . . 6 setvar 𝑦
87cv 1352 . . . . 5 class 𝑦
98, 2wcel 2148 . . . 4 wff 𝑦𝐵
106, 9wa 104 . . 3 wff (𝑥𝐴𝑦𝐵)
1110, 4, 7copab 4060 . 2 class {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝐵)}
123, 11wceq 1353 1 wff (𝐴 × 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝐵)}
Colors of variables: wff set class
This definition is referenced by:  xpeq1  4637  xpeq2  4638  elxpi  4639  elxp  4640  nfxp  4650  fconstmpt  4670  brab2a  4676  xpundi  4679  xpundir  4680  opabssxp  4697  csbxpg  4704  xpss12  4730  inxp  4757  dmxpm  4843  dmxpid  4844  resopab  4947  cnvxp  5043  xpcom  5171  dfxp3  6189  dmaddpq  7366  dmmulpq  7367  enq0enq  7418  npsspw  7458  shftfvalg  10808  shftfval  10811  reldvdsrsrg  13083  dvdsrvald  13084  dvdsrex  13089
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