Theorem cnvco 4915

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 1656	. . . 4 |- ( E. z ( x B z /\ z A y ) <-> E. z ( z A y /\ x B z ) )
2		vex 2805	. . . . 5 |- x e. _V
3		vex 2805	. . . . 5 |- y e. _V
4	2, 3	brco 4901	. . . 4 |- ( x ( A o. B ) y <-> E. z ( x B z /\ z A y ) )
5		vex 2805	. . . . . . 7 |- z e. _V
6	3, 5	brcnv 4913	. . . . . 6 |- ( y `' A z <-> z A y )
7	5, 2	brcnv 4913	. . . . . 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 1653	. . . 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 4156	. 2 |- { <. y , x >. | x ( A o. B ) y } = { <. y , x >. | E. z ( y `' A z /\ z `' B x ) }
12		df-cnv 4733	. 2 |- `' ( A o. B ) = { <. y , x >. | x ( A o. B ) y }
13		df-co 4734	. 2 |- ( `' B o. `' A ) = { <. y , x >. | E. z ( y `' A z /\ z `' B x ) }
14	11, 12, 13	3eqtr4i 2262	1 |- `' ( A o. B ) = ( `' B o. `' A )

Syntax hints:   /\ wa 104   = wceq 1397   E.wex 1540   class class class wbr 4088   {copab 4149   `'ccnv 4724   o. ccom 4729

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-br 4089  df-opab 4151  df-cnv 4733  df-co 4734

This theorem is referenced by:  rncoss  5003  rncoeq  5006  dmco  5245  cores2  5249  co01  5251  coi2  5253  relcnvtr  5256  dfdm2  5271  f1co  5554  cofunex2g  6271  caseinj  7287  djuinj  7304  cnco  14944  hmeoco  15039