Theorem cnvxp 5147

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.

Step	Hyp	Ref	Expression
1		cnvopab 5130	. . 3 |- `' { <. y , x >. | ( y e. A /\ x e. B ) } = { <. x , y >. | ( y e. A /\ x e. B ) }
2		ancom 266	. . . 4 |- ( ( y e. A /\ x e. B ) <-> ( x e. B /\ y e. A ) )
3	2	opabbii 4151	. . 3 |- { <. x , y >. | ( y e. A /\ x e. B ) } = { <. x , y >. | ( x e. B /\ y e. A ) }
4	1, 3	eqtri 2250	. 2 |- `' { <. y , x >. | ( y e. A /\ x e. B ) } = { <. x , y >. | ( x e. B /\ y e. A ) }
5		df-xp 4725	. . 3 |- ( A X. B ) = { <. y , x >. | ( y e. A /\ x e. B ) }
6	5	cnveqi 4897	. 2 |- `' ( A X. B ) = `' { <. y , x >. | ( y e. A /\ x e. B ) }
7		df-xp 4725	. 2 |- ( B X. A ) = { <. x , y >. | ( x e. B /\ y e. A ) }
8	4, 6, 7	3eqtr4i 2260	1 |- `' ( A X. B ) = ( B X. A )

Syntax hints:   /\ wa 104   = wceq 1395   e. wcel 2200   {copab 4144   X. cxp 4717   `'ccnv 4718

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-br 4084  df-opab 4146  df-xp 4725  df-rel 4726  df-cnv 4727

This theorem is referenced by:  xp0  5148  rnxpm  5158  rnxpss  5160  dminxp  5173  imainrect  5174  tposfo  6417  tposf  6418  xpider  6753  xpcomf1o  6984  pw1nct  16369