Theorem vonval 39430

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 39429	. . 3 ⊢ voln = (𝑥 ∈ Fin ↦ ((voln*‘𝑥) ↾ (CaraGen‘(voln*‘𝑥))))
2	1	a1i 11	. 2 ⊢ (𝜑 → voln = (𝑥 ∈ Fin ↦ ((voln*‘𝑥) ↾ (CaraGen‘(voln*‘𝑥)))))
3		fveq2 6103	. . . 4 ⊢ (𝑥 = 𝑋 → (voln*‘𝑥) = (voln*‘𝑋))
4	3	fveq2d 6107	. . . 4 ⊢ (𝑥 = 𝑋 → (CaraGen‘(voln*‘𝑥)) = (CaraGen‘(voln*‘𝑋)))
5	3, 4	reseq12d 5318	. . 3 ⊢ (𝑥 = 𝑋 → ((voln*‘𝑥) ↾ (CaraGen‘(voln*‘𝑥))) = ((voln*‘𝑋) ↾ (CaraGen‘(voln*‘𝑋))))
6	5	adantl 481	. 2 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → ((voln*‘𝑥) ↾ (CaraGen‘(voln*‘𝑥))) = ((voln*‘𝑋) ↾ (CaraGen‘(voln*‘𝑋))))
7		vonval.1	. 2 ⊢ (𝜑 → 𝑋 ∈ Fin)
8		fvex 6113	. . . 4 ⊢ (voln*‘𝑋) ∈ V
9	8	resex 5363	. . 3 ⊢ ((voln*‘𝑋) ↾ (CaraGen‘(voln*‘𝑋))) ∈ V
10	9	a1i 11	. 2 ⊢ (𝜑 → ((voln*‘𝑋) ↾ (CaraGen‘(voln*‘𝑋))) ∈ V)
11	2, 6, 7, 10	fvmptd 6197	1 ⊢ (𝜑 → (voln‘𝑋) = ((voln*‘𝑋) ↾ (CaraGen‘(voln*‘𝑋))))

Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ↦ cmpt 4643   ↾ cres 5040  ‘cfv 5804  Fincfn 7841  CaraGenccaragen 39381  voln*covoln 39426  volncvoln 39428

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-res 5050  df-iota 5768  df-fun 5806  df-fv 5812  df-voln 39429

This theorem is referenced by:  vonmea  39464  dmvon  39496  voncmpl  39511  mblvon  39529  vonval2  39559