Theorem vonval 46496

Description: Value of the Lebesgue measure for a given finite dimension. (Contributed by Glauco Siliprandi, 11-Oct-2020.)

Hypothesis

Ref	Expression
vonval.1	⊢ (𝜑 → 𝑋 ∈ 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.

Step	Hyp	Ref	Expression
1		df-voln 46495	. 2 ⊢ voln = (𝑥 ∈ Fin ↦ ((voln*‘𝑥) ↾ (CaraGen‘(voln*‘𝑥))))
2		fveq2 6907	. . 3 ⊢ (𝑥 = 𝑋 → (voln*‘𝑥) = (voln*‘𝑋))
3		2fveq3 6912	. . 3 ⊢ (𝑥 = 𝑋 → (CaraGen‘(voln*‘𝑥)) = (CaraGen‘(voln*‘𝑋)))
4	2, 3	reseq12d 6001	. 2 ⊢ (𝑥 = 𝑋 → ((voln*‘𝑥) ↾ (CaraGen‘(voln*‘𝑥))) = ((voln*‘𝑋) ↾ (CaraGen‘(voln*‘𝑋))))
5		vonval.1	. 2 ⊢ (𝜑 → 𝑋 ∈ Fin)
6		fvex 6920	. . . 4 ⊢ (voln*‘𝑋) ∈ V
7	6	resex 6049	. . 3 ⊢ ((voln*‘𝑋) ↾ (CaraGen‘(voln*‘𝑋))) ∈ V
8	7	a1i 11	. 2 ⊢ (𝜑 → ((voln*‘𝑋) ↾ (CaraGen‘(voln*‘𝑋))) ∈ V)
9	1, 4, 5, 8	fvmptd3 7039	1 ⊢ (𝜑 → (voln‘𝑋) = ((voln*‘𝑋) ↾ (CaraGen‘(voln*‘𝑋))))

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2106  Vcvv 3478   ↾ cres 5691  ‘cfv 6563  Fincfn 8984  CaraGenccaragen 46447  voln*covoln 46492  volncvoln 46494

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-res 5701  df-iota 6516  df-fun 6565  df-fv 6571  df-voln 46495

This theorem is referenced by:  vonmea  46530  dmvon  46562  voncmpl  46577  mblvon  46595