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Theorem dfrel3 5281
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 5274 . 2  |-  ( Rel 
R  <->  `' `' R  =  R
)
2 cnvcnv2 5277 . . 3  |-  `' `' R  =  ( R  |` 
_V )
32eqeq1i 2409 . 2  |-  ( `' `' R  =  R  <->  ( R  |`  _V )  =  R )
41, 3bitri 249 1  |-  ( Rel 
R  <->  ( R  |`  _V )  =  R
)
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    = wceq 1405   _Vcvv 3059   `'ccnv 4822    |` cres 4825   Rel wrel 4828
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pr 4630
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-rab 2763  df-v 3061  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-nul 3739  df-if 3886  df-sn 3973  df-pr 3975  df-op 3979  df-br 4396  df-opab 4454  df-xp 4829  df-rel 4830  df-cnv 4831  df-res 4835
This theorem is referenced by:  cocnvcnv2  5335  f1ovi  5835
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