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Theorem vonval 42699
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 42698 . 2 voln = (𝑥 ∈ Fin ↦ ((voln*‘𝑥) ↾ (CaraGen‘(voln*‘𝑥))))
2 fveq2 6663 . . 3 (𝑥 = 𝑋 → (voln*‘𝑥) = (voln*‘𝑋))
3 2fveq3 6668 . . 3 (𝑥 = 𝑋 → (CaraGen‘(voln*‘𝑥)) = (CaraGen‘(voln*‘𝑋)))
42, 3reseq12d 5847 . 2 (𝑥 = 𝑋 → ((voln*‘𝑥) ↾ (CaraGen‘(voln*‘𝑥))) = ((voln*‘𝑋) ↾ (CaraGen‘(voln*‘𝑋))))
5 vonval.1 . 2 (𝜑𝑋 ∈ Fin)
6 fvex 6676 . . . 4 (voln*‘𝑋) ∈ V
76resex 5892 . . 3 ((voln*‘𝑋) ↾ (CaraGen‘(voln*‘𝑋))) ∈ V
87a1i 11 . 2 (𝜑 → ((voln*‘𝑋) ↾ (CaraGen‘(voln*‘𝑋))) ∈ V)
91, 4, 5, 8fvmptd3 6783 1 (𝜑 → (voln‘𝑋) = ((voln*‘𝑋) ↾ (CaraGen‘(voln*‘𝑋))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1528  wcel 2105  Vcvv 3492  cres 5550  cfv 6348  Fincfn 8497  CaraGenccaragen 42650  voln*covoln 42695  volncvoln 42697
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-res 5560  df-iota 6307  df-fun 6350  df-fv 6356  df-voln 42698
This theorem is referenced by:  vonmea  42733  dmvon  42765  voncmpl  42780  mblvon  42798
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