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Theorem cnvxp 4685
Description: The converse of a cross product. Exercise 11 of [Suppes] p. 67. (Contributed by NM, 14-Aug-1999.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)
Assertion
Ref Expression
cnvxp  `'  X.  X.

Proof of Theorem cnvxp
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cnvopab 4669 . . 3  `' { <. ,  >.  |  }  { <. ,  >.  |  }
2 ancom 253 . . . 4
32opabbii 3815 . . 3  { <. ,  >.  |  }  { <. ,  >.  |  }
41, 3eqtri 2057 . 2  `' { <. ,  >.  |  }  { <. ,  >.  |  }
5 df-xp 4294 . . 3  X.  { <. ,  >.  |  }
65cnveqi 4453 . 2  `'  X.  `' { <. ,  >.  |  }
7 df-xp 4294 . 2  X.  { <. , 
>.  |  }
84, 6, 73eqtr4i 2067 1  `'  X.  X.
Colors of variables: wff set class
Syntax hints:   wa 97   wceq 1242   wcel 1390   {copab 3808    X. cxp 4286   `'ccnv 4287
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-bndl 1396  ax-4 1397  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
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-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-xp 4294  df-rel 4295  df-cnv 4296
This theorem is referenced by:  xp0  4686  rnxpm  4695  rnxpss  4697  dminxp  4708  imainrect  4709  tposfo  5827  tposf  5828  xpiderm  6113  xpcomf1o  6235
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