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Theorem cnvcnv3 5058
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 4617 . 2  |-  `' `' R  =  { <. x ,  y >.  |  y `' R x }
2 vex 2733 . . . 4  |-  y  e. 
_V
3 vex 2733 . . . 4  |-  x  e. 
_V
42, 3brcnv 4792 . . 3  |-  ( y `' R x  <->  x R
y )
54opabbii 4054 . 2  |-  { <. x ,  y >.  |  y `' R x }  =  { <. x ,  y
>.  |  x R
y }
61, 5eqtri 2191 1  |-  `' `' R  =  { <. x ,  y >.  |  x R y }
Colors of variables: wff set class
Syntax hints:    = wceq 1348   class class class wbr 3987   {copab 4047   `'ccnv 4608
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-br 3988  df-opab 4049  df-cnv 4617
This theorem is referenced by:  dfrel4v  5060
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