Theorem elrel 4875

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 4779	. . . 4 |- ( Rel R <-> R C_ ( _V X. _V ) )
2	1	biimpi 120	. . 3 |- ( Rel R -> R C_ ( _V X. _V ) )
3	2	sselda 3248	. 2 |- ( ( Rel R /\ A e. R ) -> A e. ( _V X. _V ) )
4		elvv 4835	. 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 1402   E.wex 1545   e. wcel 2209   _Vcvv 2821   C_ wss 3220   <.cop 3711   X. cxp 4770   Rel wrel 4777

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-pow 4309  ax-pr 4344

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-opab 4191  df-xp 4778  df-rel 4779

This theorem is referenced by:  eliunxp  4917  reldmm  4998  elres  5097  unielrel  5313  funopsn  5885  rntpos  6521