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Theorem cnvcnv3 5272
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 4830 . 2  |-  `' `' R  =  { <. x ,  y >.  |  y `' R x }
2 vex 3061 . . . 4  |-  y  e. 
_V
3 vex 3061 . . . 4  |-  x  e. 
_V
42, 3brcnv 5005 . . 3  |-  ( y `' R x  <->  x R
y )
54opabbii 4458 . 2  |-  { <. x ,  y >.  |  y `' R x }  =  { <. x ,  y
>.  |  x R
y }
61, 5eqtri 2431 1  |-  `' `' R  =  { <. x ,  y >.  |  x R y }
Colors of variables: wff setvar class
Syntax hints:    = wceq 1405   class class class wbr 4394   {copab 4451   `'ccnv 4821
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pr 4629
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-rab 2762  df-v 3060  df-dif 3416  df-un 3418  df-in 3420  df-ss 3427  df-nul 3738  df-if 3885  df-sn 3972  df-pr 3974  df-op 3978  df-br 4395  df-opab 4453  df-cnv 4830
This theorem is referenced by:  dfrel4v  5274
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