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Theorem dfrel2 5140
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 5067 . . 3  |-  Rel  `' `' R
2 vex 2804 . . . . . 6  |-  x  e. 
_V
3 vex 2804 . . . . . 6  |-  y  e. 
_V
42, 3opelcnv 4879 . . . . 5  |-  ( <.
x ,  y >.  e.  `' `' R  <->  <. y ,  x >.  e.  `' R )
53, 2opelcnv 4879 . . . . 5  |-  ( <.
y ,  x >.  e.  `' R  <->  <. x ,  y
>.  e.  R )
64, 5bitri 240 . . . 4  |-  ( <.
x ,  y >.  e.  `' `' R  <->  <. x ,  y
>.  e.  R )
76eqrelriv 4796 . . 3  |-  ( ( Rel  `' `' R  /\  Rel  R )  ->  `' `' R  =  R
)
81, 7mpan 651 . 2  |-  ( Rel 
R  ->  `' `' R  =  R )
9 releq 4787 . . 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 1632    e. wcel 1696   <.cop 3656   `'ccnv 4704   Rel wrel 4710
This theorem is referenced by:  dfrel4v  5141  cnvcnv  5142  cnveqb  5145  dfrel3  5147  cnvcnvres  5152  cnvsn  5171  cores2  5201  co01  5203  coi2  5205  relcnvtr  5208  relcnvexb  5226  funcnvres2  5339  f1cnvcnv  5461  f1ocnv  5501  f1ocnvb  5502  f1ococnv1  5518  isores1  5847  cnvf1o  6233  fnwelem  6246  tposf12  6275  ssenen  7051  cantnffval2  7413  fsumcnv  12252  structcnvcnv  13175  imasless  13458  oppcinv  13694  cnvps  14337  cnvpsb  14338  cnvtsr  14347  gimcnv  14747  lmimcnv  15836  hmeocnv  17469  hmeocnvb  17481  cmphaushmeo  17507  pi1xfrcnv  18571  dvlog  20014  efopnlem2  20020  fimacnvinrn  23214  gtiso  23256  relexprel  24046  twsymr  25181  dupre2  25347  mxlmnl2  25373  supwval  25387  f1ocan2fv  26498  ltrncnvnid  30938
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-xp 4711  df-rel 4712  df-cnv 4713
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