Theorem eldmressnsn 5651

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 4398	. 2 ⊢ (𝐴 ∈ dom 𝐹 → 𝐴 ∈ {𝐴})
2		dmressnsn 5650	. 2 ⊢ (𝐴 ∈ dom 𝐹 → dom (𝐹 ↾ {𝐴}) = {𝐴})
3	1, 2	eleqtrrd 2881	1 ⊢ (𝐴 ∈ dom 𝐹 → 𝐴 ∈ dom (𝐹 ↾ {𝐴}))

Syntax hints:   → wi 4   ∈ wcel 2157  {csn 4368  dom cdm 5312   ↾ cres 5314

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-br 4844  df-opab 4906  df-xp 5318  df-dm 5322  df-res 5324

This theorem is referenced by:  eldmeldmressn  5652  fvn0fvelrn  6658  funressndmfvrn  41927  dfdfat2  41982