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Theorem dmresv 5067
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 4910 . 2 dom (𝐴 ↾ V) = (V ∩ dom 𝐴)
2 incom 3319 . 2 (V ∩ dom 𝐴) = (dom 𝐴 ∩ V)
3 inv1 3450 . 2 (dom 𝐴 ∩ V) = dom 𝐴
41, 2, 33eqtri 2195 1 dom (𝐴 ↾ V) = dom 𝐴
Colors of variables: wff set class
Syntax hints:   = wceq 1348  Vcvv 2730  cin 3120  dom cdm 4609  cres 4611
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4105  ax-pow 4158  ax-pr 4192
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-br 3988  df-opab 4049  df-xp 4615  df-dm 4619  df-res 4621
This theorem is referenced by: (None)
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