Theorem dfrel4v 4985

Description: A relation can be expressed as the set of ordered pairs in it. (Contributed by Mario Carneiro, 16-Aug-2015.)

Assertion

Ref	Expression
dfrel4v	|- ( Rel R <-> R = { <. x , y >. | x R y } )

Distinct variable group:   x, y, R

Proof of Theorem dfrel4v

Step	Hyp	Ref	Expression
1		dfrel2 4984	. 2 |- ( Rel R <-> `' `' R = R )
2		eqcom 2139	. 2 |- ( `' `' R = R <-> R = `' `' R )
3		cnvcnv3 4983	. . 3 |- `' `' R = { <. x , y >. | x R y }
4	3	eqeq2i 2148	. 2 |- ( R = `' `' R <-> R = { <. x , y >. | x R y } )
5	1, 2, 4	3bitri 205	1 |- ( Rel R <-> R = { <. x , y >. | x R y } )

Syntax hints:   <-> wb 104   = wceq 1331   class class class wbr 3924   {copab 3983   `'ccnv 4533   Rel wrel 4539

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-br 3925  df-opab 3985  df-xp 4540  df-rel 4541  df-cnv 4542

This theorem is referenced by:  dffn5im  5460