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Theorem dfrel2 4868
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 4797 . . 3  |-  Rel  `' `' R
2 vex 2622 . . . . . 6  |-  x  e. 
_V
3 vex 2622 . . . . . 6  |-  y  e. 
_V
42, 3opelcnv 4606 . . . . 5  |-  ( <.
x ,  y >.  e.  `' `' R  <->  <. y ,  x >.  e.  `' R )
53, 2opelcnv 4606 . . . . 5  |-  ( <.
y ,  x >.  e.  `' R  <->  <. x ,  y
>.  e.  R )
64, 5bitri 182 . . . 4  |-  ( <.
x ,  y >.  e.  `' `' R  <->  <. x ,  y
>.  e.  R )
76eqrelriv 4519 . . 3  |-  ( ( Rel  `' `' R  /\  Rel  R )  ->  `' `' R  =  R
)
81, 7mpan 415 . 2  |-  ( Rel 
R  ->  `' `' R  =  R )
9 releq 4508 . . 3  |-  ( `' `' R  =  R  ->  ( Rel  `' `' R 
<->  Rel  R ) )
101, 9mpbii 146 . 2  |-  ( `' `' R  =  R  ->  Rel  R )
118, 10impbii 124 1  |-  ( Rel 
R  <->  `' `' R  =  R
)
Colors of variables: wff set class
Syntax hints:    <-> wb 103    = wceq 1289    e. wcel 1438   <.cop 3444   `'ccnv 4427   Rel wrel 4433
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3001  df-in 3003  df-ss 3010  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-br 3838  df-opab 3892  df-xp 4434  df-rel 4435  df-cnv 4436
This theorem is referenced by:  dfrel4v  4869  cnvcnv  4870  cnveqb  4873  dfrel3  4875  cnvcnvres  4881  cnvsn  4900  cores2  4930  co01  4932  coi2  4934  relcnvtr  4937  relcnvexb  4957  funcnvres2  5075  f1cnvcnv  5211  f1ocnv  5250  f1ocnvb  5251  f1ococnv1  5266  isores1  5575  cnvf1o  5972  tposf12  6016  ssenen  6547  relcnvfi  6629  fsumcnv  10794
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