Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  vonval Structured version   Visualization version   GIF version

Theorem vonval 41280
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 41279 . . 3 voln = (𝑥 ∈ Fin ↦ ((voln*‘𝑥) ↾ (CaraGen‘(voln*‘𝑥))))
21a1i 11 . 2 (𝜑 → voln = (𝑥 ∈ Fin ↦ ((voln*‘𝑥) ↾ (CaraGen‘(voln*‘𝑥)))))
3 fveq2 6348 . . . 4 (𝑥 = 𝑋 → (voln*‘𝑥) = (voln*‘𝑋))
4 2fveq3 6353 . . . 4 (𝑥 = 𝑋 → (CaraGen‘(voln*‘𝑥)) = (CaraGen‘(voln*‘𝑋)))
53, 4reseq12d 5547 . . 3 (𝑥 = 𝑋 → ((voln*‘𝑥) ↾ (CaraGen‘(voln*‘𝑥))) = ((voln*‘𝑋) ↾ (CaraGen‘(voln*‘𝑋))))
65adantl 468 . 2 ((𝜑𝑥 = 𝑋) → ((voln*‘𝑥) ↾ (CaraGen‘(voln*‘𝑥))) = ((voln*‘𝑋) ↾ (CaraGen‘(voln*‘𝑋))))
7 vonval.1 . 2 (𝜑𝑋 ∈ Fin)
8 fvex 6359 . . . 4 (voln*‘𝑋) ∈ V
98resex 5594 . . 3 ((voln*‘𝑋) ↾ (CaraGen‘(voln*‘𝑋))) ∈ V
109a1i 11 . 2 (𝜑 → ((voln*‘𝑋) ↾ (CaraGen‘(voln*‘𝑋))) ∈ V)
112, 6, 7, 10fvmptd 6447 1 (𝜑 → (voln‘𝑋) = ((voln*‘𝑋) ↾ (CaraGen‘(voln*‘𝑋))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1634  wcel 2148  Vcvv 3355  cmpt 4876  cres 5265  cfv 6042  Fincfn 8130  CaraGenccaragen 41231  voln*covoln 41276  volncvoln 41278
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1873  ax-4 1888  ax-5 1994  ax-6 2060  ax-7 2096  ax-9 2157  ax-10 2177  ax-11 2193  ax-12 2206  ax-13 2411  ax-ext 2754  ax-sep 4928  ax-nul 4936  ax-pr 5048
This theorem depends on definitions:  df-bi 198  df-an 384  df-or 864  df-3an 1100  df-tru 1637  df-ex 1856  df-nf 1861  df-sb 2053  df-eu 2625  df-mo 2626  df-clab 2761  df-cleq 2767  df-clel 2770  df-nfc 2905  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3357  df-sbc 3594  df-csb 3689  df-dif 3732  df-un 3734  df-in 3736  df-ss 3743  df-nul 4074  df-if 4236  df-sn 4327  df-pr 4329  df-op 4333  df-uni 4586  df-br 4798  df-opab 4860  df-mpt 4877  df-id 5171  df-xp 5269  df-rel 5270  df-cnv 5271  df-co 5272  df-dm 5273  df-res 5275  df-iota 6005  df-fun 6044  df-fv 6050  df-voln 41279
This theorem is referenced by:  vonmea  41314  dmvon  41346  voncmpl  41361  mblvon  41379
  Copyright terms: Public domain W3C validator