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Theorem cnvco2 30188
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 5227 . 2  |-  Rel  `' ( A  o.  `' B )
2 relco 5353 . 2  |-  Rel  ( B  o.  `' A
)
3 vex 3090 . . . . . 6  |-  y  e. 
_V
4 vex 3090 . . . . . 6  |-  z  e. 
_V
53, 4brcnv 5037 . . . . 5  |-  ( y `' B z  <->  z B
y )
6 vex 3090 . . . . . . 7  |-  x  e. 
_V
76, 4brcnv 5037 . . . . . 6  |-  ( x `' A z  <->  z A x )
87bicomi 205 . . . . 5  |-  ( z A x  <->  x `' A z )
95, 8anbi12ci 702 . . . 4  |-  ( ( y `' B z  /\  z A x )  <->  ( x `' A z  /\  z B y ) )
109exbii 1714 . . 3  |-  ( E. z ( y `' B z  /\  z A x )  <->  E. z
( x `' A
z  /\  z B
y ) )
116, 3opelcnv 5036 . . . 4  |-  ( <.
x ,  y >.  e.  `' ( A  o.  `' B )  <->  <. y ,  x >.  e.  ( A  o.  `' B
) )
123, 6opelco 5026 . . . 4  |-  ( <.
y ,  x >.  e.  ( A  o.  `' B )  <->  E. z
( y `' B
z  /\  z A x ) )
1311, 12bitri 252 . . 3  |-  ( <.
x ,  y >.  e.  `' ( A  o.  `' B )  <->  E. z
( y `' B
z  /\  z A x ) )
146, 3opelco 5026 . . 3  |-  ( <.
x ,  y >.  e.  ( B  o.  `' A )  <->  E. z
( x `' A
z  /\  z B
y ) )
1510, 13, 143bitr4i 280 . 2  |-  ( <.
x ,  y >.  e.  `' ( A  o.  `' B )  <->  <. x ,  y >.  e.  ( B  o.  `' A
) )
161, 2, 15eqrelriiv 4949 1  |-  `' ( A  o.  `' B
)  =  ( B  o.  `' A )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 370    = wceq 1437   E.wex 1659    e. wcel 1870   <.cop 4008   class class class wbr 4426   `'ccnv 4853    o. ccom 4858
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pr 4661
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-br 4427  df-opab 4485  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863
This theorem is referenced by: (None)
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