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Theorem cnvco1 24117
Description: Another distributive law of converse over class composition. (Contributed by Scott Fenton, 3-May-2014.)
Assertion
Ref Expression
cnvco1  |-  `' ( `' A  o.  B
)  =  ( `' B  o.  A )

Proof of Theorem cnvco1
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relcnv 5051 . 2  |-  Rel  `' ( `' A  o.  B
)
2 relco 5171 . 2  |-  Rel  ( `' B  o.  A
)
3 vex 2791 . . . . . . 7  |-  z  e. 
_V
4 vex 2791 . . . . . . 7  |-  y  e. 
_V
53, 4brcnv 4864 . . . . . 6  |-  ( z `' B y  <->  y B
z )
65bicomi 193 . . . . 5  |-  ( y B z  <->  z `' B y )
7 vex 2791 . . . . . 6  |-  x  e. 
_V
83, 7brcnv 4864 . . . . 5  |-  ( z `' A x  <->  x A
z )
96, 8anbi12ci 679 . . . 4  |-  ( ( y B z  /\  z `' A x )  <->  ( x A z  /\  z `' B y ) )
109exbii 1569 . . 3  |-  ( E. z ( y B z  /\  z `' A x )  <->  E. z
( x A z  /\  z `' B
y ) )
117, 4opelcnv 4863 . . . 4  |-  ( <.
x ,  y >.  e.  `' ( `' A  o.  B )  <->  <. y ,  x >.  e.  ( `' A  o.  B
) )
124, 7opelco 4853 . . . 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 ) )
147, 4opelco 4853 . . 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 4781 1  |-  `' ( `' A  o.  B
)  =  ( `' B  o.  A )
Colors of variables: wff set class
Syntax hints:    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684   <.cop 3643   class class class wbr 4023   `'ccnv 4688    o. ccom 4693
This theorem is referenced by:  pprodcnveq  24423
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698
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