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Theorem xp2 5741
 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 5739 . . 3 (x (A × B) ↔ (x (V × V) ((1stx) A (2ndx) B)))
21abbi2i 2149 . 2 (A × B) = {x ∣ (x (V × V) ((1stx) A (2ndx) B))}
3 df-rab 2309 . 2 {x (V × V) ∣ ((1stx) A (2ndx) B)} = {x ∣ (x (V × V) ((1stx) A (2ndx) B))}
42, 3eqtr4i 2060 1 (A × B) = {x (V × V) ∣ ((1stx) A (2ndx) B)}
 Colors of variables: wff set class Syntax hints:   ∧ wa 97   = wceq 1242   ∈ wcel 1390  {cab 2023  {crab 2304  Vcvv 2551   × cxp 4286  ‘cfv 4845  1st c1st 5707  2nd c2nd 5708 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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136 This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-sbc 2759  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-fo 4851  df-fv 4853  df-1st 5709  df-2nd 5710 This theorem is referenced by:  unielxp  5742
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