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Theorem dfrel2 5457
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 5374 . . 3  |-  Rel  `' `' R
2 vex 3116 . . . . . 6  |-  x  e. 
_V
3 vex 3116 . . . . . 6  |-  y  e. 
_V
42, 3opelcnv 5184 . . . . 5  |-  ( <.
x ,  y >.  e.  `' `' R  <->  <. y ,  x >.  e.  `' R )
53, 2opelcnv 5184 . . . . 5  |-  ( <.
y ,  x >.  e.  `' R  <->  <. x ,  y
>.  e.  R )
64, 5bitri 249 . . . 4  |-  ( <.
x ,  y >.  e.  `' `' R  <->  <. x ,  y
>.  e.  R )
76eqrelriv 5096 . . 3  |-  ( ( Rel  `' `' R  /\  Rel  R )  ->  `' `' R  =  R
)
81, 7mpan 670 . 2  |-  ( Rel 
R  ->  `' `' R  =  R )
9 releq 5085 . . 3  |-  ( `' `' R  =  R  ->  ( Rel  `' `' R 
<->  Rel  R ) )
101, 9mpbii 211 . 2  |-  ( `' `' R  =  R  ->  Rel  R )
118, 10impbii 188 1  |-  ( Rel 
R  <->  `' `' R  =  R
)
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    = wceq 1379    e. wcel 1767   <.cop 4033   `'ccnv 4998   Rel wrel 5004
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 4568  ax-nul 4576  ax-pr 4686
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 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-opab 4506  df-xp 5005  df-rel 5006  df-cnv 5007
This theorem is referenced by:  dfrel4v  5458  cnvcnv  5459  cnveqb  5462  dfrel3  5464  cnvcnvres  5471  cnvsn  5491  cores2  5520  co01  5522  coi2  5524  relcnvtr  5527  funcnvres2  5659  f1cnvcnv  5789  f1ocnv  5828  f1ocnvb  5829  f1ococnv1  5844  isores1  6218  relcnvexb  6732  cnvf1o  6882  fnwelem  6898  tposf12  6980  ssenen  7691  cantnffval2  8114  cantnffval2OLD  8136  fsumcnv  13551  structcnvcnv  14501  imasless  14795  oppcinv  15031  cnvps  15699  cnvpsb  15700  cnvtsr  15709  gimcnv  16120  lmimcnv  17513  hmeocnv  20026  hmeocnvb  20038  cmphaushmeo  20064  ustexsym  20481  pi1xfrcnv  21320  dvlog  22788  efopnlem2  22794  fimacnvinrn  27176  gtiso  27219  relexprel  28560  fprodcnv  28718  f1ocan2fv  29849  ltrncnvnid  34941
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