Theorem reldmresv 32402

Description: The scalar restriction is a proper operator, so it can be used with ovprc1 7435. (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 32401	. 2 ⊢ ↾v = (𝑦 ∈ V, 𝑥 ∈ V ↦ if((Base‘(Scalar‘𝑦)) ⊆ 𝑥, 𝑦, (𝑦 sSet ⟨(Scalar‘ndx), ((Scalar‘𝑦) ↾s 𝑥)⟩)))
2	1	reldmmpo 7530	1 ⊢ Rel dom ↾v

Syntax hints:  Vcvv 3475   ⊆ wss 3946  ifcif 4524  ⟨cop 4630  dom cdm 5672  Rel wrel 5677  ‘cfv 6535  (class class class)co 7396   sSet csts 17083  ndxcnx 17113  Basecbs 17131   ↾_s cress 17160  Scalarcsca 17187   ↾_v cresv 32400

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5295  ax-nul 5302  ax-pr 5423

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-rab 3434  df-v 3477  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4321  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-br 5145  df-opab 5207  df-xp 5678  df-rel 5679  df-dm 5682  df-oprab 7400  df-mpo 7401  df-resv 32401

This theorem is referenced by:  resvsca  32406  resvlem  32407  resvlemOLD  32408