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 32710
Description: The scalar restriction is a proper operator, so it can be used with ovprc1 7450. (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 32709 . 2 v = (𝑦 ∈ V, 𝑥 ∈ V ↦ if((Base‘(Scalar‘𝑦)) ⊆ 𝑥, 𝑦, (𝑦 sSet ⟨(Scalar‘ndx), ((Scalar‘𝑦) ↾s 𝑥)⟩)))
21reldmmpo 7545 1 Rel dom ↾v
Colors of variables: wff setvar class
Syntax hints:  Vcvv 3472  wss 3947  ifcif 4527  cop 4633  dom cdm 5675  Rel wrel 5680  cfv 6542  (class class class)co 7411   sSet csts 17100  ndxcnx 17130  Basecbs 17148  s cress 17177  Scalarcsca 17204  v cresv 32708
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-br 5148  df-opab 5210  df-xp 5681  df-rel 5682  df-dm 5685  df-oprab 7415  df-mpo 7416  df-resv 32709
This theorem is referenced by:  resvsca  32714  resvlem  32715  resvlemOLD  32716
  Copyright terms: Public domain W3C validator