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Theorem vonval 40082
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 40081 . . 3 voln = (𝑥 ∈ Fin ↦ ((voln*‘𝑥) ↾ (CaraGen‘(voln*‘𝑥))))
21a1i 11 . 2 (𝜑 → voln = (𝑥 ∈ Fin ↦ ((voln*‘𝑥) ↾ (CaraGen‘(voln*‘𝑥)))))
3 fveq2 6153 . . . 4 (𝑥 = 𝑋 → (voln*‘𝑥) = (voln*‘𝑋))
43fveq2d 6157 . . . 4 (𝑥 = 𝑋 → (CaraGen‘(voln*‘𝑥)) = (CaraGen‘(voln*‘𝑋)))
53, 4reseq12d 5362 . . 3 (𝑥 = 𝑋 → ((voln*‘𝑥) ↾ (CaraGen‘(voln*‘𝑥))) = ((voln*‘𝑋) ↾ (CaraGen‘(voln*‘𝑋))))
65adantl 482 . 2 ((𝜑𝑥 = 𝑋) → ((voln*‘𝑥) ↾ (CaraGen‘(voln*‘𝑥))) = ((voln*‘𝑋) ↾ (CaraGen‘(voln*‘𝑋))))
7 vonval.1 . 2 (𝜑𝑋 ∈ Fin)
8 fvex 6163 . . . 4 (voln*‘𝑋) ∈ V
98resex 5407 . . 3 ((voln*‘𝑋) ↾ (CaraGen‘(voln*‘𝑋))) ∈ V
109a1i 11 . 2 (𝜑 → ((voln*‘𝑋) ↾ (CaraGen‘(voln*‘𝑋))) ∈ V)
112, 6, 7, 10fvmptd 6250 1 (𝜑 → (voln‘𝑋) = ((voln*‘𝑋) ↾ (CaraGen‘(voln*‘𝑋))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  wcel 1987  Vcvv 3189  cmpt 4678  cres 5081  cfv 5852  Fincfn 7906  CaraGenccaragen 40033  voln*covoln 40078  volncvoln 40080
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-res 5091  df-iota 5815  df-fun 5854  df-fv 5860  df-voln 40081
This theorem is referenced by:  vonmea  40116  dmvon  40148  voncmpl  40163  mblvon  40181
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