Theorem eldmeldmressn 6026

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 𝐹 ↔ 𝑋 ∈ dom (𝐹 ↾ {𝑋}))

Proof of Theorem eldmeldmressn

Step	Hyp	Ref	Expression
1		eldmressnsn 6025	. 2 ⊢ (𝑋 ∈ dom 𝐹 → 𝑋 ∈ dom (𝐹 ↾ {𝑋}))
2		elinel2 4155	. . 3 ⊢ (𝑋 ∈ ({𝑋} ∩ dom 𝐹) → 𝑋 ∈ dom 𝐹)
3		dmres 6013	. . 3 ⊢ dom (𝐹 ↾ {𝑋}) = ({𝑋} ∩ dom 𝐹)
4	2, 3	eleq2s 2883	. 2 ⊢ (𝑋 ∈ dom (𝐹 ↾ {𝑋}) → 𝑋 ∈ dom 𝐹)
5	1, 4	impbii 212	1 ⊢ (𝑋 ∈ dom 𝐹 ↔ 𝑋 ∈ dom (𝐹 ↾ {𝑋}))

This proof depends on syntax axioms:   ↔ wb 209   ∈ wcel 2146   ∩ cin 3905  {csn 4591  dom cdm 5663   ↾ cres 5665

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737  ax-sep 5259  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-br 5112  df-opab 5176  df-xp 5669  df-dm 5673  df-res 5675

This theorem is used by: (None)