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Theorem dmresss 42774
Description: The domain of a restriction is a subset of the original domain. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
Assertion
Ref Expression
dmresss dom (𝐴𝐵) ⊆ dom 𝐴

Proof of Theorem dmresss
StepHypRef Expression
1 dmres 5913 . 2 dom (𝐴𝐵) = (𝐵 ∩ dom 𝐴)
2 inss2 4163 . 2 (𝐵 ∩ dom 𝐴) ⊆ dom 𝐴
31, 2eqsstri 3955 1 dom (𝐴𝐵) ⊆ dom 𝐴
Colors of variables: wff setvar class
Syntax hints:  cin 3886  wss 3887  dom cdm 5589  cres 5591
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-opab 5137  df-xp 5595  df-dm 5599  df-res 5601
This theorem is referenced by:  limsupresuz2  43250  liminfresuz2  43328
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