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Theorem vonval 45809
Description: Value of the Lebesgue measure for a given finite dimension. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
Hypothesis
Ref Expression
vonval.1 (πœ‘ β†’ 𝑋 ∈ Fin)
Assertion
Ref Expression
vonval (πœ‘ β†’ (volnβ€˜π‘‹) = ((voln*β€˜π‘‹) β†Ύ (CaraGenβ€˜(voln*β€˜π‘‹))))

Proof of Theorem vonval
Dummy variable π‘₯ is distinct from all other variables.
StepHypRef Expression
1 df-voln 45808 . 2 voln = (π‘₯ ∈ Fin ↦ ((voln*β€˜π‘₯) β†Ύ (CaraGenβ€˜(voln*β€˜π‘₯))))
2 fveq2 6884 . . 3 (π‘₯ = 𝑋 β†’ (voln*β€˜π‘₯) = (voln*β€˜π‘‹))
3 2fveq3 6889 . . 3 (π‘₯ = 𝑋 β†’ (CaraGenβ€˜(voln*β€˜π‘₯)) = (CaraGenβ€˜(voln*β€˜π‘‹)))
42, 3reseq12d 5975 . 2 (π‘₯ = 𝑋 β†’ ((voln*β€˜π‘₯) β†Ύ (CaraGenβ€˜(voln*β€˜π‘₯))) = ((voln*β€˜π‘‹) β†Ύ (CaraGenβ€˜(voln*β€˜π‘‹))))
5 vonval.1 . 2 (πœ‘ β†’ 𝑋 ∈ Fin)
6 fvex 6897 . . . 4 (voln*β€˜π‘‹) ∈ V
76resex 6022 . . 3 ((voln*β€˜π‘‹) β†Ύ (CaraGenβ€˜(voln*β€˜π‘‹))) ∈ V
87a1i 11 . 2 (πœ‘ β†’ ((voln*β€˜π‘‹) β†Ύ (CaraGenβ€˜(voln*β€˜π‘‹))) ∈ V)
91, 4, 5, 8fvmptd3 7014 1 (πœ‘ β†’ (volnβ€˜π‘‹) = ((voln*β€˜π‘‹) β†Ύ (CaraGenβ€˜(voln*β€˜π‘‹))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1533   ∈ wcel 2098  Vcvv 3468   β†Ύ cres 5671  β€˜cfv 6536  Fincfn 8938  CaraGenccaragen 45760  voln*covoln 45805  volncvoln 45807
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-res 5681  df-iota 6488  df-fun 6538  df-fv 6544  df-voln 45808
This theorem is referenced by:  vonmea  45843  dmvon  45875  voncmpl  45890  mblvon  45908
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