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Theorem eldmeldmressn 6045
Description: An element of the domain (of a relation) is an element of the domain of the restriction (of the relation) to the singleton containing this element. (Contributed by Alexander van der Vekens, 22-Jul-2018.)
Assertion
Ref Expression
eldmeldmressn (𝑋 ∈ dom 𝐹𝑋 ∈ dom (𝐹 ↾ {𝑋}))

Proof of Theorem eldmeldmressn
StepHypRef Expression
1 eldmressnsn 6044 . 2 (𝑋 ∈ dom 𝐹𝑋 ∈ dom (𝐹 ↾ {𝑋}))
2 elinel2 4212 . . 3 (𝑋 ∈ ({𝑋} ∩ dom 𝐹) → 𝑋 ∈ dom 𝐹)
3 dmres 6032 . . 3 dom (𝐹 ↾ {𝑋}) = ({𝑋} ∩ dom 𝐹)
42, 3eleq2s 2857 . 2 (𝑋 ∈ dom (𝐹 ↾ {𝑋}) → 𝑋 ∈ dom 𝐹)
51, 4impbii 209 1 (𝑋 ∈ dom 𝐹𝑋 ∈ dom (𝐹 ↾ {𝑋}))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wcel 2106  cin 3962  {csn 4631  dom cdm 5689  cres 5691
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-xp 5695  df-dm 5699  df-res 5701
This theorem is referenced by: (None)
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