Theorem cnvco 4520

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 1499	. . . 4 |- ( E. z ( x B z /\ z A y ) <-> E. z ( z A y /\ x B z ) )
2		vex 2560	. . . . 5 |- x e. _V
3		vex 2560	. . . . 5 |- y e. _V
4	2, 3	brco 4506	. . . 4 |- ( x ( A o. B ) y <-> E. z ( x B z /\ z A y ) )
5		vex 2560	. . . . . . 7 |- z e. _V
6	3, 5	brcnv 4518	. . . . . 6 |- ( y `' A z <-> z A y )
7	5, 2	brcnv 4518	. . . . . 6 |- ( z `' B x <-> x B z )
8	6, 7	anbi12i 433	. . . . 5 |- ( ( y `' A z /\ z `' B x ) <-> ( z A y /\ x B z ) )
9	8	exbii 1496	. . . 4 |- ( E. z ( y `' A z /\ z `' B x ) <-> E. z ( z A y /\ x B z ) )
10	1, 4, 9	3bitr4i 201	. . 3 |- ( x ( A o. B ) y <-> E. z ( y `' A z /\ z `' B x ) )
11	10	opabbii 3824	. 2 |- { <. y , x >. | x ( A o. B ) y } = { <. y , x >. | E. z ( y `' A z /\ z `' B x ) }
12		df-cnv 4353	. 2 |- `' ( A o. B ) = { <. y , x >. | x ( A o. B ) y }
13		df-co 4354	. 2 |- ( `' B o. `' A ) = { <. y , x >. | E. z ( y `' A z /\ z `' B x ) }
14	11, 12, 13	3eqtr4i 2070	1 |- `' ( A o. B ) = ( `' B o. `' A )

Syntax hints:   /\ wa 97   = wceq 1243   E.wex 1381   class class class wbr 3764   {copab 3817   `'ccnv 4344   o. ccom 4349

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-cnv 4353  df-co 4354

This theorem is referenced by:  rncoss  4602  rncoeq  4605  dmco  4829  cores2  4833  co01  4835  coi2  4837  relcnvtr  4840  dfdm2  4852  f1co  5101  cofunex2g  5739