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Theorem vonval 46555
Description: Value of the Lebesgue measure for a given finite dimension. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
Hypothesis
Ref Expression
vonval.1 (𝜑𝑋 ∈ Fin)
Assertion
Ref Expression
vonval (𝜑 → (voln‘𝑋) = ((voln*‘𝑋) ↾ (CaraGen‘(voln*‘𝑋))))

Proof of Theorem vonval
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 df-voln 46554 . 2 voln = (𝑥 ∈ Fin ↦ ((voln*‘𝑥) ↾ (CaraGen‘(voln*‘𝑥))))
2 fveq2 6906 . . 3 (𝑥 = 𝑋 → (voln*‘𝑥) = (voln*‘𝑋))
3 2fveq3 6911 . . 3 (𝑥 = 𝑋 → (CaraGen‘(voln*‘𝑥)) = (CaraGen‘(voln*‘𝑋)))
42, 3reseq12d 5998 . 2 (𝑥 = 𝑋 → ((voln*‘𝑥) ↾ (CaraGen‘(voln*‘𝑥))) = ((voln*‘𝑋) ↾ (CaraGen‘(voln*‘𝑋))))
5 vonval.1 . 2 (𝜑𝑋 ∈ Fin)
6 fvex 6919 . . . 4 (voln*‘𝑋) ∈ V
76resex 6047 . . 3 ((voln*‘𝑋) ↾ (CaraGen‘(voln*‘𝑋))) ∈ V
87a1i 11 . 2 (𝜑 → ((voln*‘𝑋) ↾ (CaraGen‘(voln*‘𝑋))) ∈ V)
91, 4, 5, 8fvmptd3 7039 1 (𝜑 → (voln‘𝑋) = ((voln*‘𝑋) ↾ (CaraGen‘(voln*‘𝑋))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  Vcvv 3480  cres 5687  cfv 6561  Fincfn 8985  CaraGenccaragen 46506  voln*covoln 46551  volncvoln 46553
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 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-res 5697  df-iota 6514  df-fun 6563  df-fv 6569  df-voln 46554
This theorem is referenced by:  vonmea  46589  dmvon  46621  voncmpl  46636  mblvon  46654
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