Theorem dmressnsn 6023

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 6003	. 2 ⊢ dom (𝐹 ↾ {𝐴}) = ({𝐴} ∩ dom 𝐹)
2		snssi 4811	. . 3 ⊢ (𝐴 ∈ dom 𝐹 → {𝐴} ⊆ dom 𝐹)
3		df-ss 3965	. . 3 ⊢ ({𝐴} ⊆ dom 𝐹 ↔ ({𝐴} ∩ dom 𝐹) = {𝐴})
4	2, 3	sylib 217	. 2 ⊢ (𝐴 ∈ dom 𝐹 → ({𝐴} ∩ dom 𝐹) = {𝐴})
5	1, 4	eqtrid 2783	1 ⊢ (𝐴 ∈ dom 𝐹 → dom (𝐹 ↾ {𝐴}) = {𝐴})

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2105   ∩ cin 3947   ⊆ wss 3948  {csn 4628  dom cdm 5676   ↾ cres 5678

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702  ax-sep 5299  ax-nul 5306  ax-pr 5427

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-xp 5682  df-dm 5686  df-res 5688

This theorem is referenced by:  eldmressnsn  6024  funcoressn  46051  funressnfv  46052