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Theorem elrel 5040
Description: A member of a relation is an ordered pair. (Contributed by NM, 17-Sep-2006.)
Assertion
Ref Expression
elrel  |-  ( ( Rel  R  /\  A  e.  R )  ->  E. x E. y  A  =  <. x ,  y >.
)
Distinct variable group:    x, y, A
Allowed substitution hints:    R( x, y)

Proof of Theorem elrel
StepHypRef Expression
1 df-rel 4945 . . . 4  |-  ( Rel 
R  <->  R  C_  ( _V 
X.  _V ) )
21biimpi 194 . . 3  |-  ( Rel 
R  ->  R  C_  ( _V  X.  _V ) )
32sselda 3454 . 2  |-  ( ( Rel  R  /\  A  e.  R )  ->  A  e.  ( _V  X.  _V ) )
4 elvv 4995 . 2  |-  ( A  e.  ( _V  X.  _V )  <->  E. x E. y  A  =  <. x ,  y >. )
53, 4sylib 196 1  |-  ( ( Rel  R  /\  A  e.  R )  ->  E. x E. y  A  =  <. x ,  y >.
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370   E.wex 1587    e. wcel 1758   _Vcvv 3068    C_ wss 3426   <.cop 3981    X. cxp 4936   Rel wrel 4943
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4511  ax-nul 4519  ax-pr 4629
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-v 3070  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-nul 3736  df-if 3890  df-sn 3976  df-pr 3978  df-op 3982  df-opab 4449  df-xp 4944  df-rel 4945
This theorem is referenced by:  eliunxp  5075  elres  5243  unielrel  5460  frxp  6782  rntpos  6858  gsum2d2lem  16570  dfpo2  27699  fundmpss  27711  sscoid  28078  elfuns  28080  eliunxp2  30859
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