Theorem cnvco2 31650

Description: Another distributive law of converse over class composition. (Contributed by Scott Fenton, 3-May-2014.)

Assertion

Ref	Expression
cnvco2	|- `' ( A o. `' B ) = ( B o. `' A )

Proof of Theorem cnvco2

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		relcnv 5503	. 2 |- Rel `' ( A o. `' B )
2		relco 5633	. 2 |- Rel ( B o. `' A )
3		vex 3203	. . . . . 6 |- y e. _V
4		vex 3203	. . . . . 6 |- z e. _V
5	3, 4	brcnv 5305	. . . . 5 |- ( y `' B z <-> z B y )
6		vex 3203	. . . . . . 7 |- x e. _V
7	6, 4	brcnv 5305	. . . . . 6 |- ( x `' A z <-> z A x )
8	7	bicomi 214	. . . . 5 |- ( z A x <-> x `' A z )
9	5, 8	anbi12ci 734	. . . 4 |- ( ( y `' B z /\ z A x ) <-> ( x `' A z /\ z B y ) )
10	9	exbii 1774	. . 3 |- ( E. z ( y `' B z /\ z A x ) <-> E. z ( x `' A z /\ z B y ) )
11	6, 3	opelcnv 5304	. . . 4 |- ( <. x , y >. e. `' ( A o. `' B ) <-> <. y , x >. e. ( A o. `' B ) )
12	3, 6	opelco 5293	. . . 4 |- ( <. y , x >. e. ( A o. `' B ) <-> E. z ( y `' B z /\ z A x ) )
13	11, 12	bitri 264	. . 3 |- ( <. x , y >. e. `' ( A o. `' B ) <-> E. z ( y `' B z /\ z A x ) )
14	6, 3	opelco 5293	. . 3 |- ( <. x , y >. e. ( B o. `' A ) <-> E. z ( x `' A z /\ z B y ) )
15	10, 13, 14	3bitr4i 292	. 2 |- ( <. x , y >. e. `' ( A o. `' B ) <-> <. x , y >. e. ( B o. `' A ) )
16	1, 2, 15	eqrelriiv 5214	1 |- `' ( A o. `' B ) = ( B o. `' A )

Syntax hints:   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   <.cop 4183   class class class wbr 4653   `'ccnv 5113   o. ccom 5118

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123

This theorem is referenced by: (None)