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Theorem dfrel4v 4960
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
StepHypRef Expression
1 dfrel2 4959 . 2  |-  ( Rel 
R  <->  `' `' R  =  R
)
2 eqcom 2119 . 2  |-  ( `' `' R  =  R  <->  R  =  `' `' R
)
3 cnvcnv3 4958 . . 3  |-  `' `' R  =  { <. x ,  y >.  |  x R y }
43eqeq2i 2128 . 2  |-  ( R  =  `' `' R  <->  R  =  { <. x ,  y >.  |  x R y } )
51, 2, 43bitri 205 1  |-  ( Rel 
R  <->  R  =  { <. x ,  y >.  |  x R y } )
Colors of variables: wff set class
Syntax hints:    <-> wb 104    = wceq 1316   class class class wbr 3899   {copab 3958   `'ccnv 4508   Rel wrel 4514
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-br 3900  df-opab 3960  df-xp 4515  df-rel 4516  df-cnv 4517
This theorem is referenced by:  dffn5im  5435
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