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Theorem dfrel3 5261
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 5254 . 2  |-  ( Rel 
R  <->  `' `' R  =  R
)
2 cnvcnv2 5257 . . 3  |-  `' `' R  =  ( R  |` 
_V )
32eqeq1i 2387 . 2  |-  ( `' `' R  =  R  <->  ( R  |`  _V )  =  R )
41, 3bitri 241 1  |-  ( Rel 
R  <->  ( R  |`  _V )  =  R
)
Colors of variables: wff set class
Syntax hints:    <-> wb 177    = wceq 1649   _Vcvv 2892   `'ccnv 4810    |` cres 4813   Rel wrel 4816
This theorem is referenced by:  cocnvcnv2  5314  f1ovi  5647
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361  ax-sep 4264  ax-nul 4272  ax-pr 4337
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2235  df-mo 2236  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-ne 2545  df-ral 2647  df-rex 2648  df-rab 2651  df-v 2894  df-dif 3259  df-un 3261  df-in 3263  df-ss 3270  df-nul 3565  df-if 3676  df-sn 3756  df-pr 3757  df-op 3759  df-br 4147  df-opab 4201  df-xp 4817  df-rel 4818  df-cnv 4819  df-res 4823
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