Theorem rellininds 49373

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 49372	. 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 5801	1 ⊢ Rel linIndS

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∀wral 3076  𝒫 cpw 4557   class class class wbr 5103  Rel wrel 5660  ‘cfv 6533  (class class class)co 7413   ↑_m cmap 8826   finSupp cfsupp 9331  Basecbs 17301  Scalarcsca 17345  0_gc0g 17524   linC clinc 49334   linIndS clininds 49370

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-v 3452  df-ss 3916  df-opab 5168  df-xp 5661  df-rel 5662  df-lininds 49372

This theorem is used by:  linindsv  49375