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Theorem cnvxp 5282
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 5266 . . 3  |-  `' { <. y ,  x >.  |  ( y  e.  A  /\  x  e.  B
) }  =  { <. x ,  y >.  |  ( y  e.  A  /\  x  e.  B ) }
2 ancom 438 . . . 4  |-  ( ( y  e.  A  /\  x  e.  B )  <->  ( x  e.  B  /\  y  e.  A )
)
32opabbii 4264 . . 3  |-  { <. x ,  y >.  |  ( y  e.  A  /\  x  e.  B ) }  =  { <. x ,  y >.  |  ( x  e.  B  /\  y  e.  A ) }
41, 3eqtri 2455 . 2  |-  `' { <. y ,  x >.  |  ( y  e.  A  /\  x  e.  B
) }  =  { <. x ,  y >.  |  ( x  e.  B  /\  y  e.  A ) }
5 df-xp 4876 . . 3  |-  ( A  X.  B )  =  { <. y ,  x >.  |  ( y  e.  A  /\  x  e.  B ) }
65cnveqi 5039 . 2  |-  `' ( A  X.  B )  =  `' { <. y ,  x >.  |  ( y  e.  A  /\  x  e.  B ) }
7 df-xp 4876 . 2  |-  ( B  X.  A )  =  { <. x ,  y
>.  |  ( x  e.  B  /\  y  e.  A ) }
84, 6, 73eqtr4i 2465 1  |-  `' ( A  X.  B )  =  ( B  X.  A )
Colors of variables: wff set class
Syntax hints:    /\ wa 359    = wceq 1652    e. wcel 1725   {copab 4257    X. cxp 4868   `'ccnv 4869
This theorem is referenced by:  xp0  5283  rnxp  5291  rnxpss  5293  dminxp  5303  imainrect  5304  fparlem3  6440  fparlem4  6441  tposfo  6498  tposf  6499  xpider  6967  xpcomf1o  7189  fpwwe2lem13  8509  xpsc  13774  pjdm  16926  ordtrest2  17260  ustneism  18245  trust  18251  metustsymOLD  18583  metustsym  18584  metustOLD  18589  metust  18590  gtiso  24080  mbfmcst  24601  0rrv  24701  elrn3  25378  xpexb  27625
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-opab 4259  df-xp 4876  df-rel 4877  df-cnv 4878
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