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Theorem cnvxp 4770
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 4754 . . 3  |-  `' { <. y ,  x >.  |  ( y  e.  A  /\  x  e.  B
) }  =  { <. x ,  y >.  |  ( y  e.  A  /\  x  e.  B ) }
2 ancom 257 . . . 4  |-  ( ( y  e.  A  /\  x  e.  B )  <->  ( x  e.  B  /\  y  e.  A )
)
32opabbii 3852 . . 3  |-  { <. x ,  y >.  |  ( y  e.  A  /\  x  e.  B ) }  =  { <. x ,  y >.  |  ( x  e.  B  /\  y  e.  A ) }
41, 3eqtri 2076 . 2  |-  `' { <. y ,  x >.  |  ( y  e.  A  /\  x  e.  B
) }  =  { <. x ,  y >.  |  ( x  e.  B  /\  y  e.  A ) }
5 df-xp 4379 . . 3  |-  ( A  X.  B )  =  { <. y ,  x >.  |  ( y  e.  A  /\  x  e.  B ) }
65cnveqi 4538 . 2  |-  `' ( A  X.  B )  =  `' { <. y ,  x >.  |  ( y  e.  A  /\  x  e.  B ) }
7 df-xp 4379 . 2  |-  ( B  X.  A )  =  { <. x ,  y
>.  |  ( x  e.  B  /\  y  e.  A ) }
84, 6, 73eqtr4i 2086 1  |-  `' ( A  X.  B )  =  ( B  X.  A )
Colors of variables: wff set class
Syntax hints:    /\ wa 101    = wceq 1259    e. wcel 1409   {copab 3845    X. cxp 4371   `'ccnv 4372
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-br 3793  df-opab 3847  df-xp 4379  df-rel 4380  df-cnv 4381
This theorem is referenced by:  xp0  4771  rnxpm  4780  rnxpss  4782  dminxp  4793  imainrect  4794  tposfo  5917  tposf  5918  xpiderm  6208  xpcomf1o  6330
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