Theorem reldmresv 33300

Description: The scalar restriction is a proper operator, so it can be used with ovprc1 7426. (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 33299	. 2 ⊢ ↾v = (𝑦 ∈ V, 𝑥 ∈ V ↦ if((Base‘(Scalar‘𝑦)) ⊆ 𝑥, 𝑦, (𝑦 sSet ⟨(Scalar‘ndx), ((Scalar‘𝑦) ↾s 𝑥)⟩)))
2	1	reldmmpo 7523	1 ⊢ Rel dom ↾v

Syntax hints:  Vcvv 3447   ⊆ wss 3914  ifcif 4488  ⟨cop 4595  dom cdm 5638  Rel wrel 5643  ‘cfv 6511  (class class class)co 7387   sSet csts 17133  ndxcnx 17163  Basecbs 17179   ↾_s cress 17200  Scalarcsca 17223   ↾_v cresv 33298

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 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-br 5108  df-opab 5170  df-xp 5644  df-rel 5645  df-dm 5648  df-oprab 7391  df-mpo 7392  df-resv 33299

This theorem is referenced by:  resvsca  33304  resvlem  33305