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Theorem cnvco1 27568
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 5204 . 2  |-  Rel  `' ( `' A  o.  B
)
2 relco 5334 . 2  |-  Rel  ( `' B  o.  A
)
3 vex 2973 . . . . . . 7  |-  z  e. 
_V
4 vex 2973 . . . . . . 7  |-  y  e. 
_V
53, 4brcnv 5020 . . . . . 6  |-  ( z `' B y  <->  y B
z )
65bicomi 202 . . . . 5  |-  ( y B z  <->  z `' B y )
7 vex 2973 . . . . . 6  |-  x  e. 
_V
83, 7brcnv 5020 . . . . 5  |-  ( z `' A x  <->  x A
z )
96, 8anbi12ci 698 . . . 4  |-  ( ( y B z  /\  z `' A x )  <->  ( x A z  /\  z `' B y ) )
109exbii 1634 . . 3  |-  ( E. z ( y B z  /\  z `' A x )  <->  E. z
( x A z  /\  z `' B
y ) )
117, 4opelcnv 5019 . . . 4  |-  ( <.
x ,  y >.  e.  `' ( `' A  o.  B )  <->  <. y ,  x >.  e.  ( `' A  o.  B
) )
124, 7opelco 5009 . . . 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 ) )
147, 4opelco 5009 . . 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 4932 1  |-  `' ( `' A  o.  B
)  =  ( `' B  o.  A )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    = wceq 1369   E.wex 1586    e. wcel 1756   <.cop 3881   class class class wbr 4290   `'ccnv 4837    o. ccom 4842
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2422  ax-sep 4411  ax-nul 4419  ax-pr 4529
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2428  df-cleq 2434  df-clel 2437  df-nfc 2566  df-ne 2606  df-ral 2718  df-rex 2719  df-rab 2722  df-v 2972  df-dif 3329  df-un 3331  df-in 3333  df-ss 3340  df-nul 3636  df-if 3790  df-sn 3876  df-pr 3878  df-op 3882  df-br 4291  df-opab 4349  df-xp 4844  df-rel 4845  df-cnv 4846  df-co 4847
This theorem is referenced by:  pprodcnveq  27912
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