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Theorem dmresv 5306
Description: The domain of a universal restriction. (Contributed by NM, 14-May-2008.)
Assertion
Ref Expression
dmresv  |-  dom  ( A  |`  _V )  =  dom  A

Proof of Theorem dmresv
StepHypRef Expression
1 dmres 5137 . 2  |-  dom  ( A  |`  _V )  =  ( _V  i^i  dom  A )
2 incom 3652 . 2  |-  ( _V 
i^i  dom  A )  =  ( dom  A  i^i  _V )
3 inv1 3786 . 2  |-  ( dom 
A  i^i  _V )  =  dom  A
41, 2, 33eqtri 2453 1  |-  dom  ( A  |`  _V )  =  dom  A
Colors of variables: wff setvar class
Syntax hints:    = wceq 1437   _Vcvv 3078    i^i cin 3432   dom cdm 4846    |` cres 4848
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4540  ax-nul 4548  ax-pr 4653
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-rab 2782  df-v 3080  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-sn 3994  df-pr 3996  df-op 4000  df-br 4418  df-opab 4477  df-xp 4852  df-dm 4856  df-res 4858
This theorem is referenced by:  fidomdm  7851  dmct  28283
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