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Theorem rellininds 47211
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 π‘š 𝑠 π‘₯ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-lininds 47210 . 2 linIndS = {βŸ¨π‘ , π‘šβŸ© ∣ (𝑠 ∈ 𝒫 (Baseβ€˜π‘š) ∧ βˆ€π‘“ ∈ ((Baseβ€˜(Scalarβ€˜π‘š)) ↑m 𝑠)((𝑓 finSupp (0gβ€˜(Scalarβ€˜π‘š)) ∧ (𝑓( linC β€˜π‘š)𝑠) = (0gβ€˜π‘š)) β†’ βˆ€π‘₯ ∈ 𝑠 (π‘“β€˜π‘₯) = (0gβ€˜(Scalarβ€˜π‘š))))}
21relopabiv 5819 1 Rel linIndS
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 394   = wceq 1539   ∈ wcel 2104  βˆ€wral 3059  π’« cpw 4601   class class class wbr 5147  Rel wrel 5680  β€˜cfv 6542  (class class class)co 7411   ↑m cmap 8822   finSupp cfsupp 9363  Basecbs 17148  Scalarcsca 17204  0gc0g 17389   linC clinc 47172   linIndS clininds 47208
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701
This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1542  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-v 3474  df-in 3954  df-ss 3964  df-opab 5210  df-xp 5681  df-rel 5682  df-lininds 47210
This theorem is referenced by:  linindsv  47213
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