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Theorem cnvxp 5362
Description: The converse of a Cartesian 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 5345 . . 3  |-  `' { <. y ,  x >.  |  ( y  e.  A  /\  x  e.  B
) }  =  { <. x ,  y >.  |  ( y  e.  A  /\  x  e.  B ) }
2 ancom 450 . . . 4  |-  ( ( y  e.  A  /\  x  e.  B )  <->  ( x  e.  B  /\  y  e.  A )
)
32opabbii 4463 . . 3  |-  { <. x ,  y >.  |  ( y  e.  A  /\  x  e.  B ) }  =  { <. x ,  y >.  |  ( x  e.  B  /\  y  e.  A ) }
41, 3eqtri 2483 . 2  |-  `' { <. y ,  x >.  |  ( y  e.  A  /\  x  e.  B
) }  =  { <. x ,  y >.  |  ( x  e.  B  /\  y  e.  A ) }
5 df-xp 4953 . . 3  |-  ( A  X.  B )  =  { <. y ,  x >.  |  ( y  e.  A  /\  x  e.  B ) }
65cnveqi 5121 . 2  |-  `' ( A  X.  B )  =  `' { <. y ,  x >.  |  ( y  e.  A  /\  x  e.  B ) }
7 df-xp 4953 . 2  |-  ( B  X.  A )  =  { <. x ,  y
>.  |  ( x  e.  B  /\  y  e.  A ) }
84, 6, 73eqtr4i 2493 1  |-  `' ( A  X.  B )  =  ( B  X.  A )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    = wceq 1370    e. wcel 1758   {copab 4456    X. cxp 4945   `'ccnv 4946
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4520  ax-nul 4528  ax-pr 4638
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2649  df-ral 2803  df-rex 2804  df-rab 2807  df-v 3078  df-dif 3438  df-un 3440  df-in 3442  df-ss 3449  df-nul 3745  df-if 3899  df-sn 3985  df-pr 3987  df-op 3991  df-br 4400  df-opab 4458  df-xp 4953  df-rel 4954  df-cnv 4955
This theorem is referenced by:  xp0  5363  rnxp  5375  rnxpss  5377  dminxp  5385  imainrect  5386  fparlem3  6783  fparlem4  6784  tposfo  6881  tposf  6882  xpider  7280  xpcomf1o  7509  fpwwe2lem13  8919  xpsc  14613  pjdm  18256  tposmap  18468  ordtrest2  18939  ustneism  19929  trust  19935  metustsymOLD  20267  metustsym  20268  metustOLD  20273  metust  20274  gtiso  26146  ordtcnvNEW  26494  ordtrest2NEW  26497  mbfmcst  26817  eulerpartlemt  26897  0rrv  26977  elrn3  27716  xpexb  29857
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