Theorem vonval 47146

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 47145	. 2 ⊢ voln = (𝑥 ∈ Fin ↦ ((voln*‘𝑥) ↾ (CaraGen‘(voln*‘𝑥))))
2		fveq2 6882	. . 3 ⊢ (𝑥 = 𝑋 → (voln*‘𝑥) = (voln*‘𝑋))
3		2fveq3 6887	. . 3 ⊢ (𝑥 = 𝑋 → (CaraGen‘(voln*‘𝑥)) = (CaraGen‘(voln*‘𝑋)))
4	2, 3	reseq12d 5980	. 2 ⊢ (𝑥 = 𝑋 → ((voln*‘𝑥) ↾ (CaraGen‘(voln*‘𝑥))) = ((voln*‘𝑋) ↾ (CaraGen‘(voln*‘𝑋))))
5		vonval.1	. 2 ⊢ (𝜑 → 𝑋 ∈ Fin)
6		fvex 6895	. . . 4 ⊢ (voln*‘𝑋) ∈ V
7	6	resex 6029	. . 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 7014	1 ⊢ (𝜑 → (voln‘𝑋) = ((voln*‘𝑋) ↾ (CaraGen‘(voln*‘𝑋))))

Syntax hints:   → wi 4   = wceq 1567   ∈ wcel 2149  Vcvv 3463   ↾ cres 5664  ‘cfv 6537  Fincfn 8943  CaraGenccaragen 47097  voln*covoln 47142  volncvoln 47144

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-res 5674  df-iota 6493  df-fun 6539  df-fv 6545  df-voln 47145

This theorem is referenced by:  vonmea  47180  dmvon  47212  voncmpl  47227  mblvon  47245