Theorem dfrel4v 5055

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 5054	. 2 |- ( Rel R <-> `' `' R = R )
2		eqcom 2167	. 2 |- ( `' `' R = R <-> R = `' `' R )
3		cnvcnv3 5053	. . 3 |- `' `' R = { <. x , y >. | x R y }
4	3	eqeq2i 2176	. 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 1343   class class class wbr 3982   {copab 4042   `'ccnv 4603   Rel wrel 4609

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-xp 4610  df-rel 4611  df-cnv 4612

This theorem is referenced by:  dffn5im  5532