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Theorem cnvcnv3 4731
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 4314 . 2  |-  `' `' R  =  { <. x ,  y >.  |  y `' R x }
2 vex 2557 . . . 4  |-  y  e. 
_V
3 vex 2557 . . . 4  |-  x  e. 
_V
42, 3brcnv 4479 . . 3  |-  ( y `' R x  <->  x R
y )
54opabbii 3820 . 2  |-  { <. x ,  y >.  |  y `' R x }  =  { <. x ,  y
>.  |  x R
y }
61, 5eqtri 2060 1  |-  `' `' R  =  { <. x ,  y >.  |  x R y }
Colors of variables: wff set class
Syntax hints:    = wceq 1243   class class class wbr 3760   {copab 3813   `'ccnv 4305
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3871  ax-pow 3923  ax-pr 3940
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2556  df-un 2919  df-in 2921  df-ss 2928  df-pw 3358  df-sn 3378  df-pr 3379  df-op 3381  df-br 3761  df-opab 3815  df-cnv 4314
This theorem is referenced by:  dfrel4v  4733
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