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Theorem dfrel4v 5315
Description: A relation can be expressed as the set of ordered pairs in it. An analogue of dffn5 5765 for relations. (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 5314 . 2  |-  ( Rel 
R  <->  `' `' R  =  R
)
2 eqcom 2438 . 2  |-  ( `' `' R  =  R  <->  R  =  `' `' R
)
3 cnvcnv3 5313 . . 3  |-  `' `' R  =  { <. x ,  y >.  |  x R y }
43eqeq2i 2446 . 2  |-  ( R  =  `' `' R  <->  R  =  { <. x ,  y >.  |  x R y } )
51, 2, 43bitri 263 1  |-  ( Rel 
R  <->  R  =  { <. x ,  y >.  |  x R y } )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    = wceq 1652   class class class wbr 4205   {copab 4258   `'ccnv 4870   Rel wrel 4876
This theorem is referenced by:  dffn5  5765  fsplit  6444  pwsle  13707  tgphaus  18139  dfrel4  24027  fneer  26360  dfafn5a  27992
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4323  ax-nul 4331  ax-pr 4396
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2703  df-rex 2704  df-rab 2707  df-v 2951  df-dif 3316  df-un 3318  df-in 3320  df-ss 3327  df-nul 3622  df-if 3733  df-sn 3813  df-pr 3814  df-op 3816  df-br 4206  df-opab 4260  df-xp 4877  df-rel 4878  df-cnv 4879
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