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Theorem elcnv 4857
Description: Membership in a converse. Equation 5 of [Suppes] p. 62. (Contributed by NM, 24-Mar-1998.)
Assertion
Ref Expression
elcnv  |-  ( A  e.  `' R  <->  E. x E. y ( A  = 
<. x ,  y >.  /\  y R x ) )
Distinct variable groups:    x, y, A    x, R, y

Proof of Theorem elcnv
StepHypRef Expression
1 df-cnv 4696 . . 3  |-  `' R  =  { <. x ,  y
>.  |  y R x }
21eleq2i 2348 . 2  |-  ( A  e.  `' R  <->  A  e.  {
<. x ,  y >.  |  y R x } )
3 elopab 4271 . 2  |-  ( A  e.  { <. x ,  y >.  |  y R x }  <->  E. x E. y ( A  = 
<. x ,  y >.  /\  y R x ) )
42, 3bitri 242 1  |-  ( A  e.  `' R  <->  E. x E. y ( A  = 
<. x ,  y >.  /\  y R x ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 178    /\ wa 360   E.wex 1529    = wceq 1624    e. wcel 1685   <.cop 3644   class class class wbr 4024   {copab 4077   `'ccnv 4687
This theorem is referenced by:  elcnv2  4858
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pr 4213
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-v 2791  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-sn 3647  df-pr 3648  df-op 3650  df-opab 4079  df-cnv 4696
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