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Theorem xp2 5799
 Description: Representation of cross product based on ordered pair component functions. (Contributed by NM, 16-Sep-2006.)
Assertion
Ref Expression
xp2 (𝐴 × 𝐵) = {𝑥 ∈ (V × V) ∣ ((1st𝑥) ∈ 𝐴 ∧ (2nd𝑥) ∈ 𝐵)}
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem xp2
StepHypRef Expression
1 elxp7 5797 . . 3 (𝑥 ∈ (𝐴 × 𝐵) ↔ (𝑥 ∈ (V × V) ∧ ((1st𝑥) ∈ 𝐴 ∧ (2nd𝑥) ∈ 𝐵)))
21abbi2i 2152 . 2 (𝐴 × 𝐵) = {𝑥 ∣ (𝑥 ∈ (V × V) ∧ ((1st𝑥) ∈ 𝐴 ∧ (2nd𝑥) ∈ 𝐵))}
3 df-rab 2315 . 2 {𝑥 ∈ (V × V) ∣ ((1st𝑥) ∈ 𝐴 ∧ (2nd𝑥) ∈ 𝐵)} = {𝑥 ∣ (𝑥 ∈ (V × V) ∧ ((1st𝑥) ∈ 𝐴 ∧ (2nd𝑥) ∈ 𝐵))}
42, 3eqtr4i 2063 1 (𝐴 × 𝐵) = {𝑥 ∈ (V × V) ∣ ((1st𝑥) ∈ 𝐴 ∧ (2nd𝑥) ∈ 𝐵)}
 Colors of variables: wff set class Syntax hints:   ∧ wa 97   = wceq 1243   ∈ wcel 1393  {cab 2026  {crab 2310  Vcvv 2557   × cxp 4343  ‘cfv 4902  1st c1st 5765  2nd c2nd 5766 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-rab 2315  df-v 2559  df-sbc 2765  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-fo 4908  df-fv 4910  df-1st 5767  df-2nd 5768 This theorem is referenced by:  unielxp  5800
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