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Theorem dfrel2 5123
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 5050 . . 3  |-  Rel  `' `' R
2 vex 2792 . . . . . 6  |-  x  e. 
_V
3 vex 2792 . . . . . 6  |-  y  e. 
_V
42, 3opelcnv 4862 . . . . 5  |-  ( <.
x ,  y >.  e.  `' `' R  <->  <. y ,  x >.  e.  `' R )
53, 2opelcnv 4862 . . . . 5  |-  ( <.
y ,  x >.  e.  `' R  <->  <. x ,  y
>.  e.  R )
64, 5bitri 240 . . . 4  |-  ( <.
x ,  y >.  e.  `' `' R  <->  <. x ,  y
>.  e.  R )
76eqrelriv 4779 . . 3  |-  ( ( Rel  `' `' R  /\  Rel  R )  ->  `' `' R  =  R
)
81, 7mpan 651 . 2  |-  ( Rel 
R  ->  `' `' R  =  R )
9 releq 4770 . . 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 1685   <.cop 3644   `'ccnv 4687   Rel wrel 4693
This theorem is referenced by:  dfrel4v  5124  cnvcnv  5125  cnveqb  5128  dfrel3  5129  cnvcnvres  5134  cnvsn  5153  cores2  5183  co01  5185  coi2  5187  relcnvtr  5190  relcnvexb  5208  funcnvres2  5289  f1cnvcnv  5411  f1ocnv  5451  f1ocnvb  5452  f1ococnv1  5468  isores1  5793  cnvf1o  6179  fnwelem  6192  tposf12  6221  ssenen  7031  cantnffval2  7393  fsumcnv  12232  structcnvcnv  13155  imasless  13438  oppcinv  13674  cnvps  14317  cnvpsb  14318  cnvtsr  14327  gimcnv  14727  lmimcnv  15816  hmeocnv  17449  hmeocnvb  17461  cmphaushmeo  17487  pi1xfrcnv  18551  dvlog  19994  efopnlem2  20000  relexprel  23438  twsymr  24488  dupre2  24655  mxlmnl2  24681  supwval  24695  f1ocan2fv  25806  ltrncnvnid  29595
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 1636  ax-8 1644  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pr 4213
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 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-sn 3647  df-pr 3648  df-op 3650  df-br 4025  df-opab 4079  df-xp 4694  df-rel 4695  df-cnv 4696
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