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Theorem dfrel4v 5464
Description: A relation can be expressed as the set of ordered pairs in it. An analogue of dffn5 5919 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 5463 . 2  |-  ( Rel 
R  <->  `' `' R  =  R
)
2 eqcom 2476 . 2  |-  ( `' `' R  =  R  <->  R  =  `' `' R
)
3 cnvcnv3 5462 . . 3  |-  `' `' R  =  { <. x ,  y >.  |  x R y }
43eqeq2i 2485 . 2  |-  ( R  =  `' `' R  <->  R  =  { <. x ,  y >.  |  x R y } )
51, 2, 43bitri 271 1  |-  ( Rel 
R  <->  R  =  { <. x ,  y >.  |  x R y } )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    = wceq 1379   class class class wbr 4453   {copab 4510   `'ccnv 5004   Rel wrel 5010
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pr 4692
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-br 4454  df-opab 4512  df-xp 5011  df-rel 5012  df-cnv 5013
This theorem is referenced by:  dffn5  5919  fsplit  6900  pwsle  14764  tgphaus  20483  dfrel4  27277  fneer  30098  dfafn5a  32035
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