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Theorem cnvxp 5274
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 5257 . . 3  |-  `' { <. y ,  x >.  |  ( y  e.  A  /\  x  e.  B
) }  =  { <. x ,  y >.  |  ( y  e.  A  /\  x  e.  B ) }
2 ancom 451 . . . 4  |-  ( ( y  e.  A  /\  x  e.  B )  <->  ( x  e.  B  /\  y  e.  A )
)
32opabbii 4490 . . 3  |-  { <. x ,  y >.  |  ( y  e.  A  /\  x  e.  B ) }  =  { <. x ,  y >.  |  ( x  e.  B  /\  y  e.  A ) }
41, 3eqtri 2458 . 2  |-  `' { <. y ,  x >.  |  ( y  e.  A  /\  x  e.  B
) }  =  { <. x ,  y >.  |  ( x  e.  B  /\  y  e.  A ) }
5 df-xp 4860 . . 3  |-  ( A  X.  B )  =  { <. y ,  x >.  |  ( y  e.  A  /\  x  e.  B ) }
65cnveqi 5029 . 2  |-  `' ( A  X.  B )  =  `' { <. y ,  x >.  |  ( y  e.  A  /\  x  e.  B ) }
7 df-xp 4860 . 2  |-  ( B  X.  A )  =  { <. x ,  y
>.  |  ( x  e.  B  /\  y  e.  A ) }
84, 6, 73eqtr4i 2468 1  |-  `' ( A  X.  B )  =  ( B  X.  A )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 370    = wceq 1437    e. wcel 1870   {copab 4483    X. cxp 4852   `'ccnv 4853
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pr 4661
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-br 4427  df-opab 4485  df-xp 4860  df-rel 4861  df-cnv 4862
This theorem is referenced by:  xp0  5275  rnxp  5287  rnxpss  5289  dminxp  5297  imainrect  5298  fparlem3  6909  fparlem4  6910  tposfo  7008  tposf  7009  xpider  7442  xpcomf1o  7667  fpwwe2lem13  9066  trclublem  13038  xpsc  15414  pjdm  19201  tposmap  19413  ordtrest2  20151  ustneism  21169  trust  21175  metustsym  21501  metust  21504  gtiso  28121  padct  28150  ordtcnvNEW  28565  ordtrest2NEW  28568  mbfmcst  28920  eulerpartlemt  29030  0rrv  29110  msrf  29968  mthmpps  30008  elrn3  30190  xpexb  36444
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