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Theorem cnvxp 4837
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  |-  `' ( A  X.  B )  =  ( B  X.  A )

Proof of Theorem cnvxp
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cnvopab 4820 . . 3  |-  `' { <. y ,  x >.  |  ( y  e.  A  /\  x  e.  B
) }  =  { <. x ,  y >.  |  ( y  e.  A  /\  x  e.  B ) }
2 ancom 262 . . . 4  |-  ( ( y  e.  A  /\  x  e.  B )  <->  ( x  e.  B  /\  y  e.  A )
)
32opabbii 3897 . . 3  |-  { <. x ,  y >.  |  ( y  e.  A  /\  x  e.  B ) }  =  { <. x ,  y >.  |  ( x  e.  B  /\  y  e.  A ) }
41, 3eqtri 2108 . 2  |-  `' { <. y ,  x >.  |  ( y  e.  A  /\  x  e.  B
) }  =  { <. x ,  y >.  |  ( x  e.  B  /\  y  e.  A ) }
5 df-xp 4434 . . 3  |-  ( A  X.  B )  =  { <. y ,  x >.  |  ( y  e.  A  /\  x  e.  B ) }
65cnveqi 4599 . 2  |-  `' ( A  X.  B )  =  `' { <. y ,  x >.  |  ( y  e.  A  /\  x  e.  B ) }
7 df-xp 4434 . 2  |-  ( B  X.  A )  =  { <. x ,  y
>.  |  ( x  e.  B  /\  y  e.  A ) }
84, 6, 73eqtr4i 2118 1  |-  `' ( A  X.  B )  =  ( B  X.  A )
Colors of variables: wff set class
Syntax hints:    /\ wa 102    = wceq 1289    e. wcel 1438   {copab 3890    X. cxp 4426   `'ccnv 4427
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-br 3838  df-opab 3892  df-xp 4434  df-rel 4435  df-cnv 4436
This theorem is referenced by:  xp0  4838  rnxpm  4847  rnxpss  4849  dminxp  4862  imainrect  4863  tposfo  6018  tposf  6019  xpiderm  6343  xpcomf1o  6521
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