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Theorem dmressnsn 6043
Description: The domain of a restriction to a singleton is a singleton. (Contributed by Alexander van der Vekens, 2-Jul-2017.)
Assertion
Ref Expression
dmressnsn (𝐴 ∈ dom 𝐹 → dom (𝐹 ↾ {𝐴}) = {𝐴})

Proof of Theorem dmressnsn
StepHypRef Expression
1 dmres 6032 . 2 dom (𝐹 ↾ {𝐴}) = ({𝐴} ∩ dom 𝐹)
2 snssi 4813 . . 3 (𝐴 ∈ dom 𝐹 → {𝐴} ⊆ dom 𝐹)
3 dfss2 3981 . . 3 ({𝐴} ⊆ dom 𝐹 ↔ ({𝐴} ∩ dom 𝐹) = {𝐴})
42, 3sylib 218 . 2 (𝐴 ∈ dom 𝐹 → ({𝐴} ∩ dom 𝐹) = {𝐴})
51, 4eqtrid 2787 1 (𝐴 ∈ dom 𝐹 → dom (𝐹 ↾ {𝐴}) = {𝐴})
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  cin 3962  wss 3963  {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:  eldmressnsn  6044  funcoressn  46992  funressnfv  46993
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