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Theorem dfrel4v 5125
Description: A relation can be expressed as the set of ordered pairs in it. An analogue of dffn5 5568 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 5124 . 2  |-  ( Rel 
R  <->  `' `' R  =  R
)
2 eqcom 2285 . 2  |-  ( `' `' R  =  R  <->  R  =  `' `' R
)
3 cnvcnv3 5123 . . 3  |-  `' `' R  =  { <. x ,  y >.  |  x R y }
43eqeq2i 2293 . 2  |-  ( R  =  `' `' R  <->  R  =  { <. x ,  y >.  |  x R y } )
51, 2, 43bitri 262 1  |-  ( Rel 
R  <->  R  =  { <. x ,  y >.  |  x R y } )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    = wceq 1623   class class class wbr 4023   {copab 4076   `'ccnv 4688   Rel wrel 4694
This theorem is referenced by:  dffn5  5568  fsplit  6223  pwsle  13391  tgphaus  17799  dfrel4  23204  fneer  26288  dfafn5a  28022
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-xp 4695  df-rel 4696  df-cnv 4697
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