Theorem cnvco 4796

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 1601	. . . 4 |- ( E. z ( x B z /\ z A y ) <-> E. z ( z A y /\ x B z ) )
2		vex 2733	. . . . 5 |- x e. _V
3		vex 2733	. . . . 5 |- y e. _V
4	2, 3	brco 4782	. . . 4 |- ( x ( A o. B ) y <-> E. z ( x B z /\ z A y ) )
5		vex 2733	. . . . . . 7 |- z e. _V
6	3, 5	brcnv 4794	. . . . . 6 |- ( y `' A z <-> z A y )
7	5, 2	brcnv 4794	. . . . . 6 |- ( z `' B x <-> x B z )
8	6, 7	anbi12i 457	. . . . 5 |- ( ( y `' A z /\ z `' B x ) <-> ( z A y /\ x B z ) )
9	8	exbii 1598	. . . 4 |- ( E. z ( y `' A z /\ z `' B x ) <-> E. z ( z A y /\ x B z ) )
10	1, 4, 9	3bitr4i 211	. . 3 |- ( x ( A o. B ) y <-> E. z ( y `' A z /\ z `' B x ) )
11	10	opabbii 4056	. 2 |- { <. y , x >. | x ( A o. B ) y } = { <. y , x >. | E. z ( y `' A z /\ z `' B x ) }
12		df-cnv 4619	. 2 |- `' ( A o. B ) = { <. y , x >. | x ( A o. B ) y }
13		df-co 4620	. 2 |- ( `' B o. `' A ) = { <. y , x >. | E. z ( y `' A z /\ z `' B x ) }
14	11, 12, 13	3eqtr4i 2201	1 |- `' ( A o. B ) = ( `' B o. `' A )

Syntax hints:   /\ wa 103   = wceq 1348   E.wex 1485   class class class wbr 3989   {copab 4049   `'ccnv 4610   o. ccom 4615

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-cnv 4619  df-co 4620

This theorem is referenced by:  rncoss  4881  rncoeq  4884  dmco  5119  cores2  5123  co01  5125  coi2  5127  relcnvtr  5130  dfdm2  5145  f1co  5415  cofunex2g  6089  caseinj  7066  djuinj  7083  cnco  13015  hmeoco  13110