Theorem reldmresv 33657

Description: The scalar restriction is a proper operator, so it can be used with ovprc1 7451. (Contributed by Thierry Arnoux, 6-Sep-2018.)

Assertion

Ref	Expression
reldmresv	⊢ Rel dom ↾v

Proof of Theorem reldmresv

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-resv 33656	. 2 ⊢ ↾v = (𝑦 ∈ V, 𝑥 ∈ V ↦ if((Base‘(Scalar‘𝑦)) ⊆ 𝑥, 𝑦, (𝑦 sSet ⟨(Scalar‘ndx), ((Scalar‘𝑦) ↾s 𝑥)⟩)))
2	1	reldmmpo 7546	1 ⊢ Rel dom ↾v

This proof depends on syntax axioms:  Vcvv 3454   ⊆ wss 3904  ifcif 4486  ⟨cop 4594  dom cdm 5660  Rel wrel 5665  ‘cfv 6536  (class class class)co 7412   sSet csts 17229  ndxcnx 17259  Basecbs 17275   ↾_s cress 17296  Scalarcsca 17319   ↾_v cresv 33655

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-10 2175  ax-11 2191  ax-12 2212  ax-ext 2734  ax-sep 5256  ax-pr 5403

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-nf 1813  df-sb 2096  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-rab 3416  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-br 5109  df-opab 5173  df-xp 5666  df-rel 5667  df-dm 5670  df-oprab 7416  df-mpo 7417  df-resv 33656

This theorem is used by:  resvsca  33661  resvlem  33662