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Theorem elcnv2 4858
Description: Membership in a converse. Equation 5 of [Suppes] p. 62. (Contributed by NM, 11-Aug-2004.)
Assertion
Ref Expression
elcnv2  |-  ( A  e.  `' R  <->  E. x E. y ( A  = 
<. x ,  y >.  /\  <. y ,  x >.  e.  R ) )
Distinct variable groups:    x, y, A    x, R, y

Proof of Theorem elcnv2
StepHypRef Expression
1 elcnv 4857 . 2  |-  ( A  e.  `' R  <->  E. x E. y ( A  = 
<. x ,  y >.  /\  y R x ) )
2 df-br 4025 . . . 4  |-  ( y R x  <->  <. y ,  x >.  e.  R
)
32anbi2i 675 . . 3  |-  ( ( A  =  <. x ,  y >.  /\  y R x )  <->  ( A  =  <. x ,  y
>.  /\  <. y ,  x >.  e.  R ) )
432exbii 1570 . 2  |-  ( E. x E. y ( A  =  <. x ,  y >.  /\  y R x )  <->  E. x E. y ( A  = 
<. x ,  y >.  /\  <. y ,  x >.  e.  R ) )
51, 4bitri 240 1  |-  ( A  e.  `' R  <->  E. x E. y ( A  = 
<. x ,  y >.  /\  <. y ,  x >.  e.  R ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1685   <.cop 3644   class class class wbr 4024   `'ccnv 4687
This theorem is referenced by:  cnvuni  4865
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1636  ax-8 1644  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pr 4213
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1631  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-br 4025  df-opab 4079  df-cnv 4696
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