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Theorem cnvcnv3 5321
Description: The set of all ordered pairs in a class is the same as the double converse. (Contributed by Mario Carneiro, 16-Aug-2015.)
Assertion
Ref Expression
cnvcnv3  |-  `' `' R  =  { <. x ,  y >.  |  x R y }
Distinct variable group:    x, y, R

Proof of Theorem cnvcnv3
StepHypRef Expression
1 df-cnv 4887 . 2  |-  `' `' R  =  { <. x ,  y >.  |  y `' R x }
2 vex 2960 . . . 4  |-  y  e. 
_V
3 vex 2960 . . . 4  |-  x  e. 
_V
42, 3brcnv 5056 . . 3  |-  ( y `' R x  <->  x R
y )
54opabbii 4273 . 2  |-  { <. x ,  y >.  |  y `' R x }  =  { <. x ,  y
>.  |  x R
y }
61, 5eqtri 2457 1  |-  `' `' R  =  { <. x ,  y >.  |  x R y }
Colors of variables: wff set class
Syntax hints:    = wceq 1653   class class class wbr 4213   {copab 4266   `'ccnv 4878
This theorem is referenced by:  dfrel4v  5323
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pr 4404
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-rab 2715  df-v 2959  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-op 3824  df-br 4214  df-opab 4268  df-cnv 4887
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