Theorem rellininds 48306

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 48305	. 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 5812	1 ⊢ Rel linIndS

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2107  ∀wral 3050  𝒫 cpw 4582   class class class wbr 5125  Rel wrel 5672  ‘cfv 6542  (class class class)co 7414   ↑_m cmap 8849   finSupp cfsupp 9384  Basecbs 17230  Scalarcsca 17280  0_gc0g 17460   linC clinc 48267   linIndS clininds 48303

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1542  df-ex 1779  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-v 3466  df-ss 3950  df-opab 5188  df-xp 5673  df-rel 5674  df-lininds 48305

This theorem is referenced by:  linindsv  48308