Theorem dmressnsn 6040

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

Assertion

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

Proof of Theorem dmressnsn

Step	Hyp	Ref	Expression
1		dmres 6029	. 2 ⊢ dom (𝐹 ↾ {𝐴}) = ({𝐴} ∩ dom 𝐹)
2		snssi 4807	. . 3 ⊢ (𝐴 ∈ dom 𝐹 → {𝐴} ⊆ dom 𝐹)
3		dfss2 3968	. . 3 ⊢ ({𝐴} ⊆ dom 𝐹 ↔ ({𝐴} ∩ dom 𝐹) = {𝐴})
4	2, 3	sylib 218	. 2 ⊢ (𝐴 ∈ dom 𝐹 → ({𝐴} ∩ dom 𝐹) = {𝐴})
5	1, 4	eqtrid 2788	1 ⊢ (𝐴 ∈ dom 𝐹 → dom (𝐹 ↾ {𝐴}) = {𝐴})

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2107   ∩ cin 3949   ⊆ wss 3950  {csn 4625  dom cdm 5684   ↾ cres 5686

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-br 5143  df-opab 5205  df-xp 5690  df-dm 5694  df-res 5696

This theorem is referenced by:  eldmressnsn  6041  funcoressn  47059  funressnfv  47060