Theorem elrel 4820

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 4725	. . . 4 |- ( Rel R <-> R C_ ( _V X. _V ) )
2	1	biimpi 120	. . 3 |- ( Rel R -> R C_ ( _V X. _V ) )
3	2	sselda 3224	. 2 |- ( ( Rel R /\ A e. R ) -> A e. ( _V X. _V ) )
4		elvv 4780	. 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 1395   E.wex 1538   e. wcel 2200   _Vcvv 2799   C_ wss 3197   <.cop 3669   X. cxp 4716   Rel wrel 4723

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-opab 4145  df-xp 4724  df-rel 4725

This theorem is referenced by:  eliunxp  4860  reldmm  4941  elres  5040  unielrel  5255  funopsn  5816  rntpos  6401