Theorem dmressnsn 6013

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 5993	. 2 ⊢ dom (𝐹 ↾ {𝐴}) = ({𝐴} ∩ dom 𝐹)
2		snssi 4803	. . 3 ⊢ (𝐴 ∈ dom 𝐹 → {𝐴} ⊆ dom 𝐹)
3		df-ss 3957	. . 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 2776	1 ⊢ (𝐴 ∈ dom 𝐹 → dom (𝐹 ↾ {𝐴}) = {𝐴})

Syntax hints:   → wi 4   = wceq 1533   ∈ wcel 2098   ∩ cin 3939   ⊆ wss 3940  {csn 4620  dom cdm 5666   ↾ cres 5668

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695  ax-sep 5289  ax-nul 5296  ax-pr 5417

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-op 4627  df-br 5139  df-opab 5201  df-xp 5672  df-dm 5676  df-res 5678

This theorem is referenced by:  eldmressnsn  6014  funcoressn  46203  funressnfv  46204