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Theorem elcnv 4846
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 4677 . . 3  |-  `' R  =  { <. x ,  y
>.  |  y R x }
21eleq2i 2322 . 2  |-  ( A  e.  `' R  <->  A  e.  {
<. x ,  y >.  |  y R x } )
3 elopab 4244 . 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 1537    = wceq 1619    e. wcel 1621   <.cop 3617   class class class wbr 3997   {copab 4050   `'ccnv 4660
This theorem is referenced by:  elcnv2  4847
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-sep 4115  ax-nul 4123  ax-pr 4186
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-v 2765  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-sn 3620  df-pr 3621  df-op 3623  df-opab 4052  df-cnv 4677
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