Theorem rellininds 48454

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 48453	. 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 5758	1 ⊢ Rel linIndS

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2110  ∀wral 3045  𝒫 cpw 4548   class class class wbr 5089  Rel wrel 5619  ‘cfv 6477  (class class class)co 7341   ↑_m cmap 8745   finSupp cfsupp 9240  Basecbs 17112  Scalarcsca 17156  0_gc0g 17335   linC clinc 48415   linIndS clininds 48451

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2112  ax-9 2120  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2722  df-clel 2804  df-v 3436  df-ss 3917  df-opab 5152  df-xp 5620  df-rel 5621  df-lininds 48453

This theorem is referenced by:  linindsv  48456