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Theorem elcnv2 5017
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 5016 . 2  |-  ( A  e.  `' R  <->  E. x E. y ( A  = 
<. x ,  y >.  /\  y R x ) )
2 df-br 4181 . . . 4  |-  ( y R x  <->  <. y ,  x >.  e.  R
)
32anbi2i 676 . . 3  |-  ( ( A  =  <. x ,  y >.  /\  y R x )  <->  ( A  =  <. x ,  y
>.  /\  <. y ,  x >.  e.  R ) )
432exbii 1590 . 2  |-  ( E. x E. y ( A  =  <. x ,  y >.  /\  y R x )  <->  E. x E. y ( A  = 
<. x ,  y >.  /\  <. y ,  x >.  e.  R ) )
51, 4bitri 241 1  |-  ( A  e.  `' R  <->  E. x E. y ( A  = 
<. x ,  y >.  /\  <. y ,  x >.  e.  R ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359   E.wex 1547    = wceq 1649    e. wcel 1721   <.cop 3785   class class class wbr 4180   `'ccnv 4844
This theorem is referenced by:  cnvuni  5024
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pr 4371
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-v 2926  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-br 4181  df-opab 4235  df-cnv 4853
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