Theorem elrel 4579

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 4484	. . . 4 |- ( Rel R <-> R C_ ( _V X. _V ) )
2	1	biimpi 119	. . 3 |- ( Rel R -> R C_ ( _V X. _V ) )
3	2	sselda 3047	. 2 |- ( ( Rel R /\ A e. R ) -> A e. ( _V X. _V ) )
4		elvv 4539	. 2 |- ( A e. ( _V X. _V ) <-> E. x E. y A = <. x , y >. )
5	3, 4	sylib 121	1 |- ( ( Rel R /\ A e. R ) -> E. x E. y A = <. x , y >. )

Syntax hints:   -> wi 4   /\ wa 103   = wceq 1299   E.wex 1436   e. wcel 1448   _Vcvv 2641   C_ wss 3021   <.cop 3477   X. cxp 4475   Rel wrel 4482

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-pow 4038  ax-pr 4069

This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-v 2643  df-un 3025  df-in 3027  df-ss 3034  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-opab 3930  df-xp 4483  df-rel 4484

This theorem is referenced by:  eliunxp  4616  elres  4791  unielrel  5002  rntpos  6084