Theorem reldmresv 33364

Description: The scalar restriction is a proper operator, so it can be used with ovprc1 7477. (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 33363	. 2 ⊢ ↾v = (𝑦 ∈ V, 𝑥 ∈ V ↦ if((Base‘(Scalar‘𝑦)) ⊆ 𝑥, 𝑦, (𝑦 sSet ⟨(Scalar‘ndx), ((Scalar‘𝑦) ↾s 𝑥)⟩)))
2	1	reldmmpo 7574	1 ⊢ Rel dom ↾v

Syntax hints:  Vcvv 3481   ⊆ wss 3966  ifcif 4534  ⟨cop 4640  dom cdm 5693  Rel wrel 5698  ‘cfv 6569  (class class class)co 7438   sSet csts 17206  ndxcnx 17236  Basecbs 17254   ↾_s cress 17283  Scalarcsca 17310   ↾_v cresv 33362

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 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5305  ax-nul 5315  ax-pr 5441

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-rab 3437  df-v 3483  df-dif 3969  df-un 3971  df-ss 3983  df-nul 4343  df-if 4535  df-sn 4635  df-pr 4637  df-op 4641  df-br 5152  df-opab 5214  df-xp 5699  df-rel 5700  df-dm 5703  df-oprab 7442  df-mpo 7443  df-resv 33363

This theorem is referenced by:  resvsca  33368  resvlem  33369  resvlemOLD  33370