Theorem eldmeldmressn 5984

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 5983	. 2 ⊢ (𝑋 ∈ dom 𝐹 → 𝑋 ∈ dom (𝐹 ↾ {𝑋}))
2		elinel2 4138	. . 3 ⊢ (𝑋 ∈ ({𝑋} ∩ dom 𝐹) → 𝑋 ∈ dom 𝐹)
3		dmres 5971	. . 3 ⊢ dom (𝐹 ↾ {𝑋}) = ({𝑋} ∩ dom 𝐹)
4	2, 3	eleq2s 2858	. 2 ⊢ (𝑋 ∈ dom (𝐹 ↾ {𝑋}) → 𝑋 ∈ dom 𝐹)
5	1, 4	impbii 210	1 ⊢ (𝑋 ∈ dom 𝐹 ↔ 𝑋 ∈ dom (𝐹 ↾ {𝑋}))

Syntax hints:   ↔ wb 207   ∈ wcel 2119   ∩ cin 3889  {csn 4562  dom cdm 5625   ↾ cres 5627

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712  ax-sep 5225  ax-pr 5369

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-br 5080  df-opab 5142  df-xp 5631  df-dm 5635  df-res 5637

This theorem is referenced by: (None)