Theorem reldmresv 32707

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 32706	. 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

Syntax hints:  Vcvv 3473   ⊆ wss 3949  ifcif 4529  ⟨cop 4635  dom cdm 5677  Rel wrel 5682  ‘cfv 6544  (class class class)co 7412   sSet csts 17101  ndxcnx 17131  Basecbs 17149   ↾_s cress 17178  Scalarcsca 17205   ↾_v cresv 32705

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-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5300  ax-nul 5307  ax-pr 5428

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-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-rab 3432  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-xp 5683  df-rel 5684  df-dm 5687  df-oprab 7416  df-mpo 7417  df-resv 32706

This theorem is referenced by:  resvsca  32711  resvlem  32712  resvlemOLD  32713