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Theorem rellininds 47212
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 47211 . 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 5820 1 Rel linIndS
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 395   = wceq 1540   ∈ wcel 2105  βˆ€wral 3060  π’« cpw 4602   class class class wbr 5148  Rel wrel 5681  β€˜cfv 6543  (class class class)co 7412   ↑m cmap 8824   finSupp cfsupp 9365  Basecbs 17149  Scalarcsca 17205  0gc0g 17390   linC clinc 47173   linIndS clininds 47209
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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1543  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-v 3475  df-in 3955  df-ss 3965  df-opab 5211  df-xp 5682  df-rel 5683  df-lininds 47211
This theorem is referenced by:  linindsv  47214
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