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Theorem dfrel4v 5263
Description: A relation can be expressed as the set of ordered pairs in it. An analogue of dffn5 5712 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 5262 . 2  |-  ( Rel 
R  <->  `' `' R  =  R
)
2 eqcom 2390 . 2  |-  ( `' `' R  =  R  <->  R  =  `' `' R
)
3 cnvcnv3 5261 . . 3  |-  `' `' R  =  { <. x ,  y >.  |  x R y }
43eqeq2i 2398 . 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 1649   class class class wbr 4154   {copab 4207   `'ccnv 4818   Rel wrel 4824
This theorem is referenced by:  dffn5  5712  fsplit  6391  pwsle  13642  tgphaus  18068  dfrel4  23878  fneer  26060  dfafn5a  27694
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pr 4345
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-rab 2659  df-v 2902  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-op 3767  df-br 4155  df-opab 4209  df-xp 4825  df-rel 4826  df-cnv 4827
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