Theorem eldmeldmressn 6009

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 6008	. 2 ⊢ (𝑋 ∈ dom 𝐹 → 𝑋 ∈ dom (𝐹 ↾ {𝑋}))
2		elinel2 4154	. . 3 ⊢ (𝑋 ∈ ({𝑋} ∩ dom 𝐹) → 𝑋 ∈ dom 𝐹)
3		dmres 5996	. . 3 ⊢ dom (𝐹 ↾ {𝑋}) = ({𝑋} ∩ dom 𝐹)
4	2, 3	eleq2s 2879	. 2 ⊢ (𝑋 ∈ dom (𝐹 ↾ {𝑋}) → 𝑋 ∈ dom 𝐹)
5	1, 4	impbii 211	1 ⊢ (𝑋 ∈ dom 𝐹 ↔ 𝑋 ∈ dom (𝐹 ↾ {𝑋}))

Syntax hints:   ↔ wb 208   ∈ wcel 2141   ∩ cin 3903  {csn 4581  dom cdm 5645   ↾ cres 5647

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733  ax-sep 5245  ax-pr 5389

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-sn 4582  df-pr 4584  df-op 4588  df-br 5100  df-opab 5162  df-xp 5651  df-dm 5655  df-res 5657

This theorem is referenced by: (None)