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Definition df-xp 4419
Description: Define the cross product of two classes. 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 are defined in using the cross-product ( Z × N ) ; the left- and right-hand sides of the cross-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 4411 . 2 class (𝐴 × 𝐵)
4 vx . . . . . 6 setvar 𝑥
54cv 1286 . . . . 5 class 𝑥
65, 1wcel 1436 . . . 4 wff 𝑥𝐴
7 vy . . . . . 6 setvar 𝑦
87cv 1286 . . . . 5 class 𝑦
98, 2wcel 1436 . . . 4 wff 𝑦𝐵
106, 9wa 102 . . 3 wff (𝑥𝐴𝑦𝐵)
1110, 4, 7copab 3875 . 2 class {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝐵)}
123, 11wceq 1287 1 wff (𝐴 × 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝐵)}
Colors of variables: wff set class
This definition is referenced by:  xpeq1  4427  xpeq2  4428  elxpi  4429  elxp  4430  nfxp  4439  fconstmpt  4455  brab2a  4461  xpundi  4464  xpundir  4465  opabssxp  4482  csbxpg  4489  xpss12  4515  inxp  4540  dmxpm  4626  resopab  4725  cnvxp  4818  xpcom  4945  dfxp3  5923  dmaddpq  6885  dmmulpq  6886  enq0enq  6937  npsspw  6977  shftfvalg  10152  shftfval  10155
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