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Theorem cnvco1 28752
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 5365 . 2  |-  Rel  `' ( `' A  o.  B
)
2 relco 5496 . 2  |-  Rel  ( `' B  o.  A
)
3 vex 3109 . . . . . . 7  |-  z  e. 
_V
4 vex 3109 . . . . . . 7  |-  y  e. 
_V
53, 4brcnv 5176 . . . . . 6  |-  ( z `' B y  <->  y B
z )
65bicomi 202 . . . . 5  |-  ( y B z  <->  z `' B y )
7 vex 3109 . . . . . 6  |-  x  e. 
_V
83, 7brcnv 5176 . . . . 5  |-  ( z `' A x  <->  x A
z )
96, 8anbi12ci 698 . . . 4  |-  ( ( y B z  /\  z `' A x )  <->  ( x A z  /\  z `' B y ) )
109exbii 1639 . . 3  |-  ( E. z ( y B z  /\  z `' A x )  <->  E. z
( x A z  /\  z `' B
y ) )
117, 4opelcnv 5175 . . . 4  |-  ( <.
x ,  y >.  e.  `' ( `' A  o.  B )  <->  <. y ,  x >.  e.  ( `' A  o.  B
) )
124, 7opelco 5165 . . . 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 5165 . . 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 5088 1  |-  `' ( `' A  o.  B
)  =  ( `' B  o.  A )
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    = wceq 1374   E.wex 1591    e. wcel 1762   <.cop 4026   class class class wbr 4440   `'ccnv 4991    o. ccom 4996
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pr 4679
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-sn 4021  df-pr 4023  df-op 4027  df-br 4441  df-opab 4499  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001
This theorem is referenced by:  pprodcnveq  29096
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