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Theorem dmresv 5317
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 5147 . 2  |-  dom  ( A  |`  _V )  =  ( _V  i^i  dom  A )
2 incom 3637 . 2  |-  ( _V 
i^i  dom  A )  =  ( dom  A  i^i  _V )
3 inv1 3773 . 2  |-  ( dom 
A  i^i  _V )  =  dom  A
41, 2, 33eqtri 2488 1  |-  dom  ( A  |`  _V )  =  dom  A
Colors of variables: wff setvar class
Syntax hints:    = wceq 1455   _Vcvv 3057    i^i cin 3415   dom cdm 4856    |` cres 4858
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-9 1907  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4541  ax-nul 4550  ax-pr 4656
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-ne 2635  df-ral 2754  df-rex 2755  df-rab 2758  df-v 3059  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3744  df-if 3894  df-sn 3981  df-pr 3983  df-op 3987  df-br 4419  df-opab 4478  df-xp 4862  df-dm 4866  df-res 4868
This theorem is referenced by:  fidomdm  7884  dmct  28350
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