Metamath Proof Explorer


Definition df-xp

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 } X. { 2 , 7 } ) = ( { <. 1 , 2 >. , <. 1 , 7 >. } u. { <. 5 , 2 >. , <. 5 , 7 >. } ) ( ex-xp ). Another example is that the set of rational numbers is defined in df-q using the Cartesian product ( ZZ X. NN ) ; 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)

Ref Expression
Assertion df-xp ( 𝐴 × 𝐵 ) = { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑥𝐴𝑦𝐵 ) }

Detailed syntax breakdown

Step Hyp Ref Expression
0 cA 𝐴
1 cB 𝐵
2 0 1 cxp ( 𝐴 × 𝐵 )
3 vx 𝑥
4 vy 𝑦
5 3 cv 𝑥
6 5 0 wcel 𝑥𝐴
7 4 cv 𝑦
8 7 1 wcel 𝑦𝐵
9 6 8 wa ( 𝑥𝐴𝑦𝐵 )
10 9 3 4 copab { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑥𝐴𝑦𝐵 ) }
11 2 10 wceq ( 𝐴 × 𝐵 ) = { ⟨ 𝑥 , 𝑦 ⟩ ∣ ( 𝑥𝐴𝑦𝐵 ) }