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Theorem dfrel2 4799
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 4731 . . 3  |-  Rel  `' `' R
2 vex 2577 . . . . . 6  |-  x  e. 
_V
3 vex 2577 . . . . . 6  |-  y  e. 
_V
42, 3opelcnv 4545 . . . . 5  |-  ( <.
x ,  y >.  e.  `' `' R  <->  <. y ,  x >.  e.  `' R )
53, 2opelcnv 4545 . . . . 5  |-  ( <.
y ,  x >.  e.  `' R  <->  <. x ,  y
>.  e.  R )
64, 5bitri 177 . . . 4  |-  ( <.
x ,  y >.  e.  `' `' R  <->  <. x ,  y
>.  e.  R )
76eqrelriv 4461 . . 3  |-  ( ( Rel  `' `' R  /\  Rel  R )  ->  `' `' R  =  R
)
81, 7mpan 408 . 2  |-  ( Rel 
R  ->  `' `' R  =  R )
9 releq 4450 . . 3  |-  ( `' `' R  =  R  ->  ( Rel  `' `' R 
<->  Rel  R ) )
101, 9mpbii 140 . 2  |-  ( `' `' R  =  R  ->  Rel  R )
118, 10impbii 121 1  |-  ( Rel 
R  <->  `' `' R  =  R
)
Colors of variables: wff set class
Syntax hints:    <-> wb 102    = wceq 1259    e. wcel 1409   <.cop 3406   `'ccnv 4372   Rel wrel 4378
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955  ax-pr 3972
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-un 2950  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-br 3793  df-opab 3847  df-xp 4379  df-rel 4380  df-cnv 4381
This theorem is referenced by:  dfrel4v  4800  cnvcnv  4801  cnveqb  4804  dfrel3  4806  cnvcnvres  4812  cnvsn  4831  cores2  4861  co01  4863  coi2  4865  relcnvtr  4868  relcnvexb  4885  funcnvres2  5002  f1cnvcnv  5128  f1ocnv  5167  f1ocnvb  5168  f1ococnv1  5183  isores1  5482  cnvf1o  5874  tposf12  5915
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