Theorem rellininds 48919

Description: The class defining the relation between a module and its linearly independent subsets is a relation. (Contributed by AV, 13-Apr-2019.)

Assertion

Ref	Expression
rellininds	⊢ Rel linIndS

Proof of Theorem rellininds

Dummy variables 𝑓 𝑚 𝑠 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-lininds 48918	. 2 ⊢ linIndS = {⟨𝑠, 𝑚⟩ ∣ (𝑠 ∈ 𝒫 (Base‘𝑚) ∧ ∀𝑓 ∈ ((Base‘(Scalar‘𝑚)) ↑m 𝑠)((𝑓 finSupp (0g‘(Scalar‘𝑚)) ∧ (𝑓( linC ‘𝑚)𝑠) = (0g‘𝑚)) → ∀𝑥 ∈ 𝑠 (𝑓‘𝑥) = (0g‘(Scalar‘𝑚))))}
2	1	relopabiv 5776	1 ⊢ Rel linIndS

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  ∀wral 3051  𝒫 cpw 4541   class class class wbr 5085  Rel wrel 5636  ‘cfv 6498  (class class class)co 7367   ↑_m cmap 8773   finSupp cfsupp 9274  Basecbs 17179  Scalarcsca 17223  0_gc0g 17402   linC clinc 48880   linIndS clininds 48916

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-v 3431  df-ss 3906  df-opab 5148  df-xp 5637  df-rel 5638  df-lininds 48918

This theorem is referenced by:  linindsv  48921