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Theorem cnvco1 25134
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 5175 . 2  |-  Rel  `' ( `' A  o.  B
)
2 relco 5301 . 2  |-  Rel  ( `' B  o.  A
)
3 vex 2895 . . . . . . 7  |-  z  e. 
_V
4 vex 2895 . . . . . . 7  |-  y  e. 
_V
53, 4brcnv 4988 . . . . . 6  |-  ( z `' B y  <->  y B
z )
65bicomi 194 . . . . 5  |-  ( y B z  <->  z `' B y )
7 vex 2895 . . . . . 6  |-  x  e. 
_V
83, 7brcnv 4988 . . . . 5  |-  ( z `' A x  <->  x A
z )
96, 8anbi12ci 680 . . . 4  |-  ( ( y B z  /\  z `' A x )  <->  ( x A z  /\  z `' B y ) )
109exbii 1589 . . 3  |-  ( E. z ( y B z  /\  z `' A x )  <->  E. z
( x A z  /\  z `' B
y ) )
117, 4opelcnv 4987 . . . 4  |-  ( <.
x ,  y >.  e.  `' ( `' A  o.  B )  <->  <. y ,  x >.  e.  ( `' A  o.  B
) )
124, 7opelco 4977 . . . 4  |-  ( <.
y ,  x >.  e.  ( `' A  o.  B )  <->  E. z
( y B z  /\  z `' A x ) )
1311, 12bitri 241 . . 3  |-  ( <.
x ,  y >.  e.  `' ( `' A  o.  B )  <->  E. z
( y B z  /\  z `' A x ) )
147, 4opelco 4977 . . 3  |-  ( <.
x ,  y >.  e.  ( `' B  o.  A )  <->  E. z
( x A z  /\  z `' B
y ) )
1510, 13, 143bitr4i 269 . 2  |-  ( <.
x ,  y >.  e.  `' ( `' A  o.  B )  <->  <. x ,  y >.  e.  ( `' B  o.  A
) )
161, 2, 15eqrelriiv 4903 1  |-  `' ( `' A  o.  B
)  =  ( `' B  o.  A )
Colors of variables: wff set class
Syntax hints:    /\ wa 359   E.wex 1547    = wceq 1649    e. wcel 1717   <.cop 3753   class class class wbr 4146   `'ccnv 4810    o. ccom 4815
This theorem is referenced by:  pprodcnveq  25440
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pr 4337
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-op 3759  df-br 4147  df-opab 4201  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820
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