Theorem dfrel4v 5135

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 5134	. 2 |- ( Rel R <-> `' `' R = R )
2		eqcom 2207	. 2 |- ( `' `' R = R <-> R = `' `' R )
3		cnvcnv3 5133	. . 3 |- `' `' R = { <. x , y >. | x R y }
4	3	eqeq2i 2216	. 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 1373   class class class wbr 4045   {copab 4105   `'ccnv 4675   Rel wrel 4681

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-14 2179  ax-ext 2187  ax-sep 4163  ax-pow 4219  ax-pr 4254

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-v 2774  df-un 3170  df-in 3172  df-ss 3179  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-br 4046  df-opab 4107  df-xp 4682  df-rel 4683  df-cnv 4684

This theorem is referenced by:  dffn5im  5626