Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  reldmresv Structured version   Visualization version   GIF version

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.
StepHypRef 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)⟩)))
21reldmmpo 7530 1 Rel dom ↾v
Colors of variables: wff setvar class
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
  Copyright terms: Public domain W3C validator