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Theorem dfrel2 5322
Description: Alternate definition of relation. Exercise 2 of [TakeutiZaring] p. 25. (Contributed by NM, 29-Dec-1996.)
Assertion
Ref Expression
dfrel2  |-  ( Rel 
R  <->  `' `' R  =  R
)

Proof of Theorem dfrel2
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relcnv 5243 . . 3  |-  Rel  `' `' R
2 vex 2960 . . . . . 6  |-  x  e. 
_V
3 vex 2960 . . . . . 6  |-  y  e. 
_V
42, 3opelcnv 5055 . . . . 5  |-  ( <.
x ,  y >.  e.  `' `' R  <->  <. y ,  x >.  e.  `' R )
53, 2opelcnv 5055 . . . . 5  |-  ( <.
y ,  x >.  e.  `' R  <->  <. x ,  y
>.  e.  R )
64, 5bitri 242 . . . 4  |-  ( <.
x ,  y >.  e.  `' `' R  <->  <. x ,  y
>.  e.  R )
76eqrelriv 4970 . . 3  |-  ( ( Rel  `' `' R  /\  Rel  R )  ->  `' `' R  =  R
)
81, 7mpan 653 . 2  |-  ( Rel 
R  ->  `' `' R  =  R )
9 releq 4960 . . 3  |-  ( `' `' R  =  R  ->  ( Rel  `' `' R 
<->  Rel  R ) )
101, 9mpbii 204 . 2  |-  ( `' `' R  =  R  ->  Rel  R )
118, 10impbii 182 1  |-  ( Rel 
R  <->  `' `' R  =  R
)
Colors of variables: wff set class
Syntax hints:    <-> wb 178    = wceq 1653    e. wcel 1726   <.cop 3818   `'ccnv 4878   Rel wrel 4884
This theorem is referenced by:  dfrel4v  5323  cnvcnv  5324  cnveqb  5327  dfrel3  5329  cnvcnvres  5334  cnvsn  5353  cores2  5383  co01  5385  coi2  5387  relcnvtr  5390  relcnvexb  5408  funcnvres2  5525  f1cnvcnv  5648  f1ocnv  5688  f1ocnvb  5689  f1ococnv1  5705  isores1  6055  cnvf1o  6446  fnwelem  6462  tposf12  6505  ssenen  7282  cantnffval2  7652  fsumcnv  12558  structcnvcnv  13481  imasless  13766  oppcinv  14002  cnvps  14645  cnvpsb  14646  cnvtsr  14655  gimcnv  15055  lmimcnv  16140  hmeocnv  17795  hmeocnvb  17807  cmphaushmeo  17833  ustexsym  18246  pi1xfrcnv  19083  dvlog  20543  efopnlem2  20549  fimacnvinrn  24048  gtiso  24089  relexprel  25135  fprodcnv  25308  f1ocan2fv  26430  ltrncnvnid  30925
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2418  ax-sep 4331  ax-nul 4339  ax-pr 4404
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-op 3824  df-br 4214  df-opab 4268  df-xp 4885  df-rel 4886  df-cnv 4887
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