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Theorem dmresv 4889
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 4734 . 2 dom (𝐴 ↾ V) = (V ∩ dom 𝐴)
2 incom 3192 . 2 (V ∩ dom 𝐴) = (dom 𝐴 ∩ V)
3 inv1 3319 . 2 (dom 𝐴 ∩ V) = dom 𝐴
41, 2, 33eqtri 2112 1 dom (𝐴 ↾ V) = dom 𝐴
Colors of variables: wff set class
Syntax hints:   = wceq 1289  Vcvv 2619  cin 2998  dom cdm 4438  cres 4440
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-br 3846  df-opab 3900  df-xp 4444  df-dm 4448  df-res 4450
This theorem is referenced by: (None)
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