Theorem dmressnsn 5983

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 5972	. 2 ⊢ dom (𝐹 ↾ {𝐴}) = ({𝐴} ∩ dom 𝐹)
2		snssi 4765	. . 3 ⊢ (𝐴 ∈ dom 𝐹 → {𝐴} ⊆ dom 𝐹)
3		dfss2 3920	. . 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 2784	1 ⊢ (𝐴 ∈ dom 𝐹 → dom (𝐹 ↾ {𝐴}) = {𝐴})

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114   ∩ cin 3901   ⊆ wss 3902  {csn 4581  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 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5242  ax-nul 5252  ax-pr 5378

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3062  df-rab 3401  df-v 3443  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4287  df-if 4481  df-sn 4582  df-pr 4584  df-op 4588  df-br 5100  df-opab 5162  df-xp 5631  df-dm 5635  df-res 5637

This theorem is referenced by:  eldmressnsn  5984  funcoressn  47324  funressnfv  47325