Theorem elrel 4834

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

Step	Hyp	Ref	Expression
1		df-rel 4738	. . . 4 |- ( Rel R <-> R C_ ( _V X. _V ) )
2	1	biimpi 120	. . 3 |- ( Rel R -> R C_ ( _V X. _V ) )
3	2	sselda 3228	. 2 |- ( ( Rel R /\ A e. R ) -> A e. ( _V X. _V ) )
4		elvv 4794	. 2 |- ( A e. ( _V X. _V ) <-> E. x E. y A = <. x , y >. )
5	3, 4	sylib 122	1 |- ( ( Rel R /\ A e. R ) -> E. x E. y A = <. x , y >. )

Syntax hints:   -> wi 4   /\ wa 104   = wceq 1398   E.wex 1541   e. wcel 2202   _Vcvv 2803   C_ wss 3201   <.cop 3676   X. cxp 4729   Rel wrel 4736

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-opab 4156  df-xp 4737  df-rel 4738

This theorem is referenced by:  eliunxp  4875  reldmm  4956  elres  5055  unielrel  5271  funopsn  5838  rntpos  6466