Theorem rellininds 44491

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 44490	. 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	relopabi 5689	1 ⊢ Rel linIndS

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1533   ∈ wcel 2110  ∀wral 3138  𝒫 cpw 4539   class class class wbr 5059  Rel wrel 5555  ‘cfv 6350  (class class class)co 7150   ↑_m cmap 8400   finSupp cfsupp 8827  Basecbs 16477  Scalarcsca 16562  0_gc0g 16707   linC clinc 44452   linIndS clininds 44488

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-rab 3147  df-v 3497  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4562  df-pr 4564  df-op 4568  df-opab 5122  df-xp 5556  df-rel 5557  df-lininds 44490

This theorem is referenced by:  linindsv  44493