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Theorem vonval 39430
 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 39429 . . 3 voln = (𝑥 ∈ Fin ↦ ((voln*‘𝑥) ↾ (CaraGen‘(voln*‘𝑥))))
21a1i 11 . 2 (𝜑 → voln = (𝑥 ∈ Fin ↦ ((voln*‘𝑥) ↾ (CaraGen‘(voln*‘𝑥)))))
3 fveq2 6103 . . . 4 (𝑥 = 𝑋 → (voln*‘𝑥) = (voln*‘𝑋))
43fveq2d 6107 . . . 4 (𝑥 = 𝑋 → (CaraGen‘(voln*‘𝑥)) = (CaraGen‘(voln*‘𝑋)))
53, 4reseq12d 5318 . . 3 (𝑥 = 𝑋 → ((voln*‘𝑥) ↾ (CaraGen‘(voln*‘𝑥))) = ((voln*‘𝑋) ↾ (CaraGen‘(voln*‘𝑋))))
65adantl 481 . 2 ((𝜑𝑥 = 𝑋) → ((voln*‘𝑥) ↾ (CaraGen‘(voln*‘𝑥))) = ((voln*‘𝑋) ↾ (CaraGen‘(voln*‘𝑋))))
7 vonval.1 . 2 (𝜑𝑋 ∈ Fin)
8 fvex 6113 . . . 4 (voln*‘𝑋) ∈ V
98resex 5363 . . 3 ((voln*‘𝑋) ↾ (CaraGen‘(voln*‘𝑋))) ∈ V
109a1i 11 . 2 (𝜑 → ((voln*‘𝑋) ↾ (CaraGen‘(voln*‘𝑋))) ∈ V)
112, 6, 7, 10fvmptd 6197 1 (𝜑 → (voln‘𝑋) = ((voln*‘𝑋) ↾ (CaraGen‘(voln*‘𝑋))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ↦ cmpt 4643   ↾ cres 5040  ‘cfv 5804  Fincfn 7841  CaraGenccaragen 39381  voln*covoln 39426  volncvoln 39428 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-res 5050  df-iota 5768  df-fun 5806  df-fv 5812  df-voln 39429 This theorem is referenced by:  vonmea  39464  dmvon  39496  voncmpl  39511  mblvon  39529  vonval2  39559
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