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Theorem elrel 4442
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 4352 . . . 4  |-  ( Rel 
R  <->  R  C_  ( _V 
X.  _V ) )
21biimpi 113 . . 3  |-  ( Rel 
R  ->  R  C_  ( _V  X.  _V ) )
32sselda 2945 . 2  |-  ( ( Rel  R  /\  A  e.  R )  ->  A  e.  ( _V  X.  _V ) )
4 elvv 4402 . 2  |-  ( A  e.  ( _V  X.  _V )  <->  E. x E. y  A  =  <. x ,  y >. )
53, 4sylib 127 1  |-  ( ( Rel  R  /\  A  e.  R )  ->  E. x E. y  A  =  <. x ,  y >.
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 97    = wceq 1243   E.wex 1381    e. wcel 1393   _Vcvv 2557    C_ wss 2917   <.cop 3378    X. cxp 4343   Rel wrel 4350
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944
This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-opab 3819  df-xp 4351  df-rel 4352
This theorem is referenced by:  eliunxp  4475  elres  4646  unielrel  4845  rntpos  5872
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