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Theorem eldmeldmressn 5889
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 5888 . 2 (𝑋 ∈ dom 𝐹𝑋 ∈ dom (𝐹 ↾ {𝑋}))
2 elinel2 4170 . . 3 (𝑋 ∈ ({𝑋} ∩ dom 𝐹) → 𝑋 ∈ dom 𝐹)
3 dmres 5868 . . 3 dom (𝐹 ↾ {𝑋}) = ({𝑋} ∩ dom 𝐹)
42, 3eleq2s 2928 . 2 (𝑋 ∈ dom (𝐹 ↾ {𝑋}) → 𝑋 ∈ dom 𝐹)
51, 4impbii 210 1 (𝑋 ∈ dom 𝐹𝑋 ∈ dom (𝐹 ↾ {𝑋}))
Colors of variables: wff setvar class
Syntax hints:  wb 207  wcel 2105  cin 3932  {csn 4557  dom cdm 5548  cres 5550
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-br 5058  df-opab 5120  df-xp 5554  df-dm 5558  df-res 5560
This theorem is referenced by: (None)
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