MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dfrel2 Structured version   Unicode version

Theorem dfrel2 5283
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 5201 . . 3  |-  Rel  `' `' R
2 vex 2970 . . . . . 6  |-  x  e. 
_V
3 vex 2970 . . . . . 6  |-  y  e. 
_V
42, 3opelcnv 5016 . . . . 5  |-  ( <.
x ,  y >.  e.  `' `' R  <->  <. y ,  x >.  e.  `' R )
53, 2opelcnv 5016 . . . . 5  |-  ( <.
y ,  x >.  e.  `' R  <->  <. x ,  y
>.  e.  R )
64, 5bitri 249 . . . 4  |-  ( <.
x ,  y >.  e.  `' `' R  <->  <. x ,  y
>.  e.  R )
76eqrelriv 4928 . . 3  |-  ( ( Rel  `' `' R  /\  Rel  R )  ->  `' `' R  =  R
)
81, 7mpan 670 . 2  |-  ( Rel 
R  ->  `' `' R  =  R )
9 releq 4917 . . 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 1369    e. wcel 1756   <.cop 3878   `'ccnv 4834   Rel wrel 4840
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pr 4526
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-rab 2719  df-v 2969  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-sn 3873  df-pr 3875  df-op 3879  df-br 4288  df-opab 4346  df-xp 4841  df-rel 4842  df-cnv 4843
This theorem is referenced by:  dfrel4v  5284  cnvcnv  5285  cnveqb  5288  dfrel3  5290  cnvcnvres  5297  cnvsn  5317  cores2  5345  co01  5347  coi2  5349  relcnvtr  5352  funcnvres2  5484  f1cnvcnv  5609  f1ocnv  5648  f1ocnvb  5649  f1ococnv1  5664  isores1  6020  relcnvexb  6521  cnvf1o  6666  fnwelem  6682  tposf12  6765  ssenen  7477  cantnffval2  7895  cantnffval2OLD  7917  fsumcnv  13232  structcnvcnv  14177  imasless  14470  oppcinv  14706  cnvps  15374  cnvpsb  15375  cnvtsr  15384  gimcnv  15786  lmimcnv  17125  hmeocnv  19310  hmeocnvb  19322  cmphaushmeo  19348  ustexsym  19765  pi1xfrcnv  20604  dvlog  22071  efopnlem2  22077  fimacnvinrn  25903  gtiso  25947  relexprel  27287  fprodcnv  27445  f1ocan2fv  28574  ltrncnvnid  33611
  Copyright terms: Public domain W3C validator