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Theorem eldmeldmressn 5360
 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 5359 . 2 (𝑋 ∈ dom 𝐹𝑋 ∈ dom (𝐹 ↾ {𝑋}))
2 elinel2 3762 . . 3 (𝑋 ∈ ({𝑋} ∩ dom 𝐹) → 𝑋 ∈ dom 𝐹)
3 dmres 5339 . . 3 dom (𝐹 ↾ {𝑋}) = ({𝑋} ∩ dom 𝐹)
42, 3eleq2s 2706 . 2 (𝑋 ∈ dom (𝐹 ↾ {𝑋}) → 𝑋 ∈ dom 𝐹)
51, 4impbii 198 1 (𝑋 ∈ dom 𝐹𝑋 ∈ dom (𝐹 ↾ {𝑋}))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ∈ wcel 1977   ∩ cin 3539  {csn 4125  dom cdm 5038   ↾ cres 5040 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-dm 5048  df-res 5050 This theorem is referenced by: (None)
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