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Theorem dfrel2 5124
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 5051 . . 3  |-  Rel  `' `' R
2 vex 2791 . . . . . 6  |-  x  e. 
_V
3 vex 2791 . . . . . 6  |-  y  e. 
_V
42, 3opelcnv 4863 . . . . 5  |-  ( <.
x ,  y >.  e.  `' `' R  <->  <. y ,  x >.  e.  `' R )
53, 2opelcnv 4863 . . . . 5  |-  ( <.
y ,  x >.  e.  `' R  <->  <. x ,  y
>.  e.  R )
64, 5bitri 240 . . . 4  |-  ( <.
x ,  y >.  e.  `' `' R  <->  <. x ,  y
>.  e.  R )
76eqrelriv 4780 . . 3  |-  ( ( Rel  `' `' R  /\  Rel  R )  ->  `' `' R  =  R
)
81, 7mpan 651 . 2  |-  ( Rel 
R  ->  `' `' R  =  R )
9 releq 4771 . . 3  |-  ( `' `' R  =  R  ->  ( Rel  `' `' R 
<->  Rel  R ) )
101, 9mpbii 202 . 2  |-  ( `' `' R  =  R  ->  Rel  R )
118, 10impbii 180 1  |-  ( Rel 
R  <->  `' `' R  =  R
)
Colors of variables: wff set class
Syntax hints:    <-> wb 176    = wceq 1623    e. wcel 1684   <.cop 3643   `'ccnv 4688   Rel wrel 4694
This theorem is referenced by:  dfrel4v  5125  cnvcnv  5126  cnveqb  5129  dfrel3  5131  cnvcnvres  5136  cnvsn  5155  cores2  5185  co01  5187  coi2  5189  relcnvtr  5192  relcnvexb  5210  funcnvres2  5323  f1cnvcnv  5445  f1ocnv  5485  f1ocnvb  5486  f1ococnv1  5502  isores1  5831  cnvf1o  6217  fnwelem  6230  tposf12  6259  ssenen  7035  cantnffval2  7397  fsumcnv  12236  structcnvcnv  13159  imasless  13442  oppcinv  13678  cnvps  14321  cnvpsb  14322  cnvtsr  14331  gimcnv  14731  lmimcnv  15820  hmeocnv  17453  hmeocnvb  17465  cmphaushmeo  17491  pi1xfrcnv  18555  dvlog  19998  efopnlem2  20004  fimacnvinrn  23199  gtiso  23241  relexprel  24031  twsymr  25078  dupre2  25244  mxlmnl2  25270  supwval  25284  f1ocan2fv  26395  ltrncnvnid  30316
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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