Theorem cnvco 4847

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 1619	. . . 4 |- ( E. z ( x B z /\ z A y ) <-> E. z ( z A y /\ x B z ) )
2		vex 2763	. . . . 5 |- x e. _V
3		vex 2763	. . . . 5 |- y e. _V
4	2, 3	brco 4833	. . . 4 |- ( x ( A o. B ) y <-> E. z ( x B z /\ z A y ) )
5		vex 2763	. . . . . . 7 |- z e. _V
6	3, 5	brcnv 4845	. . . . . 6 |- ( y `' A z <-> z A y )
7	5, 2	brcnv 4845	. . . . . 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 1616	. . . 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 4096	. 2 |- { <. y , x >. | x ( A o. B ) y } = { <. y , x >. | E. z ( y `' A z /\ z `' B x ) }
12		df-cnv 4667	. 2 |- `' ( A o. B ) = { <. y , x >. | x ( A o. B ) y }
13		df-co 4668	. 2 |- ( `' B o. `' A ) = { <. y , x >. | E. z ( y `' A z /\ z `' B x ) }
14	11, 12, 13	3eqtr4i 2224	1 |- `' ( A o. B ) = ( `' B o. `' A )

Syntax hints:   /\ wa 104   = wceq 1364   E.wex 1503   class class class wbr 4029   {copab 4089   `'ccnv 4658   o. ccom 4663

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-br 4030  df-opab 4091  df-cnv 4667  df-co 4668

This theorem is referenced by:  rncoss  4932  rncoeq  4935  dmco  5174  cores2  5178  co01  5180  coi2  5182  relcnvtr  5185  dfdm2  5200  f1co  5471  cofunex2g  6162  caseinj  7148  djuinj  7165  cnco  14389  hmeoco  14484