Theorem dfrel4v 5214

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 5213	. 2 |- ( Rel R <-> `' `' R = R )
2		eqcom 2234	. 2 |- ( `' `' R = R <-> R = `' `' R )
3		cnvcnv3 5212	. . 3 |- `' `' R = { <. x , y >. | x R y }
4	3	eqeq2i 2243	. 2 |- ( R = `' `' R <-> R = { <. x , y >. | x R y } )
5	1, 2, 4	3bitri 206	1 |- ( Rel R <-> R = { <. x , y >. | x R y } )

Syntax hints:   <-> wb 105   = wceq 1398   class class class wbr 4109   {copab 4170   `'ccnv 4748   Rel wrel 4754

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-pow 4287  ax-pr 4322

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-v 2815  df-un 3215  df-in 3217  df-ss 3224  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-br 4110  df-opab 4172  df-xp 4755  df-rel 4756  df-cnv 4757

This theorem is referenced by:  dffn5im  5722