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Theorem cnvco2 29043
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 5374 . 2  |-  Rel  `' ( A  o.  `' B )
2 relco 5505 . 2  |-  Rel  ( B  o.  `' A
)
3 vex 3116 . . . . . 6  |-  y  e. 
_V
4 vex 3116 . . . . . 6  |-  z  e. 
_V
53, 4brcnv 5185 . . . . 5  |-  ( y `' B z  <->  z B
y )
6 vex 3116 . . . . . . 7  |-  x  e. 
_V
76, 4brcnv 5185 . . . . . 6  |-  ( x `' A z  <->  z A x )
87bicomi 202 . . . . 5  |-  ( z A x  <->  x `' A z )
95, 8anbi12ci 698 . . . 4  |-  ( ( y `' B z  /\  z A x )  <->  ( x `' A z  /\  z B y ) )
109exbii 1644 . . 3  |-  ( E. z ( y `' B z  /\  z A x )  <->  E. z
( x `' A
z  /\  z B
y ) )
116, 3opelcnv 5184 . . . 4  |-  ( <.
x ,  y >.  e.  `' ( A  o.  `' B )  <->  <. y ,  x >.  e.  ( A  o.  `' B
) )
123, 6opelco 5174 . . . 4  |-  ( <.
y ,  x >.  e.  ( A  o.  `' B )  <->  E. z
( y `' B
z  /\  z A x ) )
1311, 12bitri 249 . . 3  |-  ( <.
x ,  y >.  e.  `' ( A  o.  `' B )  <->  E. z
( y `' B
z  /\  z A x ) )
146, 3opelco 5174 . . 3  |-  ( <.
x ,  y >.  e.  ( B  o.  `' A )  <->  E. z
( x `' A
z  /\  z B
y ) )
1510, 13, 143bitr4i 277 . 2  |-  ( <.
x ,  y >.  e.  `' ( A  o.  `' B )  <->  <. x ,  y >.  e.  ( B  o.  `' A
) )
161, 2, 15eqrelriiv 5097 1  |-  `' ( A  o.  `' B
)  =  ( B  o.  `' A )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    = wceq 1379   E.wex 1596    e. wcel 1767   <.cop 4033   class class class wbr 4447   `'ccnv 4998    o. ccom 5003
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-opab 4506  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008
This theorem is referenced by: (None)
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