Theorem cnvco 5198

Description: Distributive law of converse over class composition. Theorem 26 of [Suppes] p. 64. (Contributed by NM, 19-Mar-1998.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Assertion

Ref	Expression
cnvco	|- `' ( A o. B ) = ( `' B o. `' A )

Proof of Theorem cnvco

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		exancom 1672	. . . 4 |- ( E. z ( x B z /\ z A y ) <-> E. z ( z A y /\ x B z ) )
2		vex 3112	. . . . 5 |- x e. _V
3		vex 3112	. . . . 5 |- y e. _V
4	2, 3	brco 5183	. . . 4 |- ( x ( A o. B ) y <-> E. z ( x B z /\ z A y ) )
5		vex 3112	. . . . . . 7 |- z e. _V
6	3, 5	brcnv 5195	. . . . . 6 |- ( y `' A z <-> z A y )
7	5, 2	brcnv 5195	. . . . . 6 |- ( z `' B x <-> x B z )
8	6, 7	anbi12i 697	. . . . 5 |- ( ( y `' A z /\ z `' B x ) <-> ( z A y /\ x B z ) )
9	8	exbii 1668	. . . 4 |- ( E. z ( y `' A z /\ z `' B x ) <-> E. z ( z A y /\ x B z ) )
10	1, 4, 9	3bitr4i 277	. . 3 |- ( x ( A o. B ) y <-> E. z ( y `' A z /\ z `' B x ) )
11	10	opabbii 4521	. 2 |- { <. y , x >. | x ( A o. B ) y } = { <. y , x >. | E. z ( y `' A z /\ z `' B x ) }
12		df-cnv 5016	. 2 |- `' ( A o. B ) = { <. y , x >. | x ( A o. B ) y }
13		df-co 5017	. 2 |- ( `' B o. `' A ) = { <. y , x >. | E. z ( y `' A z /\ z `' B x ) }
14	11, 12, 13	3eqtr4i 2496	1 |- `' ( A o. B ) = ( `' B o. `' A )

Syntax hints:   /\ wa 369   = wceq 1395   E.wex 1613   class class class wbr 4456   {copab 4514   `'ccnv 5007   o. ccom 5012

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-br 4457  df-opab 4516  df-cnv 5016  df-co 5017

This theorem is referenced by:  rncoss  5273  rncoeq  5276  dmco  5521  cores2  5526  co01  5528  coi2  5530  relcnvtr  5533  dfdm2  5545  f1co  5796  cofunex2g  6764  fparlem3  6901  fparlem4  6902  supp0cosupp0  6957  imacosupp  6958  fsuppcolem  7878  mapfienOLD  8155  cnvps  15969  gimco  16443  gsumval3OLD  17035  gsumzf1o  17044  gsumzf1oOLD  17047  cnco  19894  ptrescn  20266  qtopcn  20341  hmeoco  20399  cncombf  22191  deg1val  22622  deg1valOLD  22623  fcoinver  27602  ofpreima  27661  mbfmco  28408  eulerpartlemmf  28511  cvmliftmolem1  28923  cvmlift2lem9a  28945  cvmlift2lem9  28953  mclsppslem  29140  relexpcnv  29260  relexpaddg  29278  ftc1anclem3  30297  trlcocnv  36589  tendoicl  36665  cdlemk45  36816  cnvtrrel  37942