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Theorem dfrel3 5082
Description: Alternate definition of relation. (Contributed by NM, 14-May-2008.)
Assertion
Ref Expression
dfrel3  |-  ( Rel 
R  <->  ( R  |`  _V )  =  R
)

Proof of Theorem dfrel3
StepHypRef Expression
1 dfrel2 5075 . 2  |-  ( Rel 
R  <->  `' `' R  =  R
)
2 cnvcnv2 5078 . . 3  |-  `' `' R  =  ( R  |` 
_V )
32eqeq1i 2185 . 2  |-  ( `' `' R  =  R  <->  ( R  |`  _V )  =  R )
41, 3bitri 184 1  |-  ( Rel 
R  <->  ( R  |`  _V )  =  R
)
Colors of variables: wff set class
Syntax hints:    <-> wb 105    = wceq 1353   _Vcvv 2737   `'ccnv 4622    |` cres 4625   Rel wrel 4628
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4206
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-br 4001  df-opab 4062  df-xp 4629  df-rel 4630  df-cnv 4631  df-res 4635
This theorem is referenced by:  cocnvcnv2  5136  f1ovi  5496
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