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Theorem vonval 44855
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 44854 . 2 voln = (π‘₯ ∈ Fin ↦ ((voln*β€˜π‘₯) β†Ύ (CaraGenβ€˜(voln*β€˜π‘₯))))
2 fveq2 6847 . . 3 (π‘₯ = 𝑋 β†’ (voln*β€˜π‘₯) = (voln*β€˜π‘‹))
3 2fveq3 6852 . . 3 (π‘₯ = 𝑋 β†’ (CaraGenβ€˜(voln*β€˜π‘₯)) = (CaraGenβ€˜(voln*β€˜π‘‹)))
42, 3reseq12d 5943 . 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 6860 . . . 4 (voln*β€˜π‘‹) ∈ V
76resex 5990 . . 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 6976 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 1542   ∈ wcel 2107  Vcvv 3448   β†Ύ cres 5640  β€˜cfv 6501  Fincfn 8890  CaraGenccaragen 44806  voln*covoln 44851  volncvoln 44853
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-res 5650  df-iota 6453  df-fun 6503  df-fv 6509  df-voln 44854
This theorem is referenced by:  vonmea  44889  dmvon  44921  voncmpl  44936  mblvon  44954
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