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Theorem dmresexg 4662
Description: The domain of a restriction to a set exists. (Contributed by NM, 7-Apr-1995.)
Assertion
Ref Expression
dmresexg (𝐵𝑉 → dom (𝐴𝐵) ∈ V)

Proof of Theorem dmresexg
StepHypRef Expression
1 dmres 4660 . 2 dom (𝐴𝐵) = (𝐵 ∩ dom 𝐴)
2 inex1g 3922 . 2 (𝐵𝑉 → (𝐵 ∩ dom 𝐴) ∈ V)
31, 2syl5eqel 2166 1 (𝐵𝑉 → dom (𝐴𝐵) ∈ V)
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 1434  Vcvv 2602  cin 2973  dom cdm 4371  cres 4373
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-br 3794  df-opab 3848  df-xp 4377  df-dm 4381  df-res 4383
This theorem is referenced by:  resfunexg  5414  resfunexgALT  5768
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