Theorem cnvco 4852

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 1622	. . . 4 |- ( E. z ( x B z /\ z A y ) <-> E. z ( z A y /\ x B z ) )
2		vex 2766	. . . . 5 |- x e. _V
3		vex 2766	. . . . 5 |- y e. _V
4	2, 3	brco 4838	. . . 4 |- ( x ( A o. B ) y <-> E. z ( x B z /\ z A y ) )
5		vex 2766	. . . . . . 7 |- z e. _V
6	3, 5	brcnv 4850	. . . . . 6 |- ( y `' A z <-> z A y )
7	5, 2	brcnv 4850	. . . . . 6 |- ( z `' B x <-> x B z )
8	6, 7	anbi12i 460	. . . . 5 |- ( ( y `' A z /\ z `' B x ) <-> ( z A y /\ x B z ) )
9	8	exbii 1619	. . . 4 |- ( E. z ( y `' A z /\ z `' B x ) <-> E. z ( z A y /\ x B z ) )
10	1, 4, 9	3bitr4i 212	. . 3 |- ( x ( A o. B ) y <-> E. z ( y `' A z /\ z `' B x ) )
11	10	opabbii 4101	. 2 |- { <. y , x >. | x ( A o. B ) y } = { <. y , x >. | E. z ( y `' A z /\ z `' B x ) }
12		df-cnv 4672	. 2 |- `' ( A o. B ) = { <. y , x >. | x ( A o. B ) y }
13		df-co 4673	. 2 |- ( `' B o. `' A ) = { <. y , x >. | E. z ( y `' A z /\ z `' B x ) }
14	11, 12, 13	3eqtr4i 2227	1 |- `' ( A o. B ) = ( `' B o. `' A )

Syntax hints:   /\ wa 104   = wceq 1364   E.wex 1506   class class class wbr 4034   {copab 4094   `'ccnv 4663   o. ccom 4668

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-pow 4208  ax-pr 4243

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-br 4035  df-opab 4096  df-cnv 4672  df-co 4673

This theorem is referenced by:  rncoss  4937  rncoeq  4940  dmco  5179  cores2  5183  co01  5185  coi2  5187  relcnvtr  5190  dfdm2  5205  f1co  5478  cofunex2g  6176  caseinj  7164  djuinj  7181  cnco  14541  hmeoco  14636