Theorem eldmressnsn 6016

Description: The element of the domain of a restriction to a singleton is the element of the singleton. (Contributed by Alexander van der Vekens, 2-Jul-2017.)

Assertion

Ref	Expression
eldmressnsn	⊢ (𝐴 ∈ dom 𝐹 → 𝐴 ∈ dom (𝐹 ↾ {𝐴}))

Proof of Theorem eldmressnsn

Step	Hyp	Ref	Expression
1		snidg 4641	. 2 ⊢ (𝐴 ∈ dom 𝐹 → 𝐴 ∈ {𝐴})
2		dmressnsn 6015	. 2 ⊢ (𝐴 ∈ dom 𝐹 → dom (𝐹 ↾ {𝐴}) = {𝐴})
3	1, 2	eleqtrrd 2838	1 ⊢ (𝐴 ∈ dom 𝐹 → 𝐴 ∈ dom (𝐹 ↾ {𝐴}))

Syntax hints:   → wi 4   ∈ wcel 2109  {csn 4606  dom cdm 5659   ↾ cres 5661

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-br 5125  df-opab 5187  df-xp 5665  df-dm 5669  df-res 5671

This theorem is referenced by:  eldmeldmressn  6017  fvn0fvelrnOLD  7158  funressndmfvrn  47040  dfdfat2  47124