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Theorem elcnv 5172
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 5002 . . 3  |-  `' R  =  { <. x ,  y
>.  |  y R x }
21eleq2i 2540 . 2  |-  ( A  e.  `' R  <->  A  e.  {
<. x ,  y >.  |  y R x } )
3 elopab 4750 . 2  |-  ( A  e.  { <. x ,  y >.  |  y R x }  <->  E. x E. y ( A  = 
<. x ,  y >.  /\  y R x ) )
42, 3bitri 249 1  |-  ( A  e.  `' R  <->  E. x E. y ( A  = 
<. x ,  y >.  /\  y R x ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1374   E.wex 1591    e. wcel 1762   <.cop 4028   class class class wbr 4442   {copab 4499   `'ccnv 4993
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pr 4681
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-v 3110  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-sn 4023  df-pr 4025  df-op 4029  df-opab 4501  df-cnv 5002
This theorem is referenced by:  elcnv2  5173  gsummpt2co  27422
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