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Theorem cnvco2 24189
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.
StepHypRef Expression
1 relcnv 5067 . 2  |-  Rel  `' ( A  o.  `' B )
2 relco 5187 . 2  |-  Rel  ( B  o.  `' A
)
3 vex 2804 . . . . . 6  |-  y  e. 
_V
4 vex 2804 . . . . . 6  |-  z  e. 
_V
53, 4brcnv 4880 . . . . 5  |-  ( y `' B z  <->  z B
y )
6 vex 2804 . . . . . . 7  |-  x  e. 
_V
76, 4brcnv 4880 . . . . . 6  |-  ( x `' A z  <->  z A x )
87bicomi 193 . . . . 5  |-  ( z A x  <->  x `' A z )
95, 8anbi12ci 679 . . . 4  |-  ( ( y `' B z  /\  z A x )  <->  ( x `' A z  /\  z B y ) )
109exbii 1572 . . 3  |-  ( E. z ( y `' B z  /\  z A x )  <->  E. z
( x `' A
z  /\  z B
y ) )
116, 3opelcnv 4879 . . . 4  |-  ( <.
x ,  y >.  e.  `' ( A  o.  `' B )  <->  <. y ,  x >.  e.  ( A  o.  `' B
) )
123, 6opelco 4869 . . . 4  |-  ( <.
y ,  x >.  e.  ( A  o.  `' B )  <->  E. z
( y `' B
z  /\  z A x ) )
1311, 12bitri 240 . . 3  |-  ( <.
x ,  y >.  e.  `' ( A  o.  `' B )  <->  E. z
( y `' B
z  /\  z A x ) )
146, 3opelco 4869 . . 3  |-  ( <.
x ,  y >.  e.  ( B  o.  `' A )  <->  E. z
( x `' A
z  /\  z B
y ) )
1510, 13, 143bitr4i 268 . 2  |-  ( <.
x ,  y >.  e.  `' ( A  o.  `' B )  <->  <. x ,  y >.  e.  ( B  o.  `' A
) )
161, 2, 15eqrelriiv 4797 1  |-  `' ( A  o.  `' B
)  =  ( B  o.  `' A )
Colors of variables: wff set class
Syntax hints:    /\ wa 358   E.wex 1531    = wceq 1632    e. wcel 1696   <.cop 3656   class class class wbr 4039   `'ccnv 4704    o. ccom 4709
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714
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