Theorem fvvolioof 43530

Description: The function value of the Lebesgue measure of an open interval composed with a function. (Contributed by Glauco Siliprandi, 3-Mar-2021.)

Hypotheses

Ref	Expression
fvvolioof.f	⊢ (𝜑 → 𝐹:𝐴⟶(ℝ* × ℝ*))
fvvolioof.x	⊢ (𝜑 → 𝑋 ∈ 𝐴)

Assertion

Ref	Expression
fvvolioof	⊢ (𝜑 → (((vol ∘ (,)) ∘ 𝐹)‘𝑋) = (vol‘((1st ‘(𝐹‘𝑋))(,)(2nd ‘(𝐹‘𝑋)))))

Proof of Theorem fvvolioof

Step	Hyp	Ref	Expression
1		fvvolioof.f	. . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶(ℝ* × ℝ*))
2	1	ffund 6604	. . 3 ⊢ (𝜑 → Fun 𝐹)
3		fvvolioof.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
4	1	fdmd 6611	. . . . 5 ⊢ (𝜑 → dom 𝐹 = 𝐴)
5	4	eqcomd 2744	. . . 4 ⊢ (𝜑 → 𝐴 = dom 𝐹)
6	3, 5	eleqtrd 2841	. . 3 ⊢ (𝜑 → 𝑋 ∈ dom 𝐹)
7		fvco 6866	. . 3 ⊢ ((Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹) → (((vol ∘ (,)) ∘ 𝐹)‘𝑋) = ((vol ∘ (,))‘(𝐹‘𝑋)))
8	2, 6, 7	syl2anc 584	. 2 ⊢ (𝜑 → (((vol ∘ (,)) ∘ 𝐹)‘𝑋) = ((vol ∘ (,))‘(𝐹‘𝑋)))
9		ioof 13179	. . . . 5 ⊢ (,):(ℝ* × ℝ*)⟶𝒫 ℝ
10		ffun 6603	. . . . 5 ⊢ ((,):(ℝ* × ℝ*)⟶𝒫 ℝ → Fun (,))
11	9, 10	ax-mp 5	. . . 4 ⊢ Fun (,)
12	11	a1i 11	. . 3 ⊢ (𝜑 → Fun (,))
13	1, 3	ffvelrnd 6962	. . . 4 ⊢ (𝜑 → (𝐹‘𝑋) ∈ (ℝ* × ℝ*))
14	9	fdmi 6612	. . . 4 ⊢ dom (,) = (ℝ* × ℝ*)
15	13, 14	eleqtrrdi 2850	. . 3 ⊢ (𝜑 → (𝐹‘𝑋) ∈ dom (,))
16		fvco 6866	. . 3 ⊢ ((Fun (,) ∧ (𝐹‘𝑋) ∈ dom (,)) → ((vol ∘ (,))‘(𝐹‘𝑋)) = (vol‘((,)‘(𝐹‘𝑋))))
17	12, 15, 16	syl2anc 584	. 2 ⊢ (𝜑 → ((vol ∘ (,))‘(𝐹‘𝑋)) = (vol‘((,)‘(𝐹‘𝑋))))
18		df-ov 7278	. . . . 5 ⊢ ((1st ‘(𝐹‘𝑋))(,)(2nd ‘(𝐹‘𝑋))) = ((,)‘⟨(1st ‘(𝐹‘𝑋)), (2nd ‘(𝐹‘𝑋))⟩)
19	18	a1i 11	. . . 4 ⊢ (𝜑 → ((1st ‘(𝐹‘𝑋))(,)(2nd ‘(𝐹‘𝑋))) = ((,)‘⟨(1st ‘(𝐹‘𝑋)), (2nd ‘(𝐹‘𝑋))⟩))
20		1st2nd2 7870	. . . . . . 7 ⊢ ((𝐹‘𝑋) ∈ (ℝ* × ℝ*) → (𝐹‘𝑋) = ⟨(1st ‘(𝐹‘𝑋)), (2nd ‘(𝐹‘𝑋))⟩)
21	13, 20	syl 17	. . . . . 6 ⊢ (𝜑 → (𝐹‘𝑋) = ⟨(1st ‘(𝐹‘𝑋)), (2nd ‘(𝐹‘𝑋))⟩)
22	21	eqcomd 2744	. . . . 5 ⊢ (𝜑 → ⟨(1st ‘(𝐹‘𝑋)), (2nd ‘(𝐹‘𝑋))⟩ = (𝐹‘𝑋))
23	22	fveq2d 6778	. . . 4 ⊢ (𝜑 → ((,)‘⟨(1st ‘(𝐹‘𝑋)), (2nd ‘(𝐹‘𝑋))⟩) = ((,)‘(𝐹‘𝑋)))
24	19, 23	eqtr2d 2779	. . 3 ⊢ (𝜑 → ((,)‘(𝐹‘𝑋)) = ((1st ‘(𝐹‘𝑋))(,)(2nd ‘(𝐹‘𝑋))))
25	24	fveq2d 6778	. 2 ⊢ (𝜑 → (vol‘((,)‘(𝐹‘𝑋))) = (vol‘((1st ‘(𝐹‘𝑋))(,)(2nd ‘(𝐹‘𝑋)))))
26	8, 17, 25	3eqtrd 2782	1 ⊢ (𝜑 → (((vol ∘ (,)) ∘ 𝐹)‘𝑋) = (vol‘((1st ‘(𝐹‘𝑋))(,)(2nd ‘(𝐹‘𝑋)))))

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2106  𝒫 cpw 4533  ⟨cop 4567   × cxp 5587  dom cdm 5589   ∘ ccom 5593  Fun wfun 6427  ⟶wf 6429  ‘cfv 6433  (class class class)co 7275  1^st c1st 7829  2^nd c2nd 7830  ℝcr 10870  ℝ^*cxr 11008  (,)cioo 13079  volcvol 24627

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pow 5288  ax-pr 5352  ax-un 7588  ax-cnex 10927  ax-resscn 10928  ax-pre-lttri 10945  ax-pre-lttrn 10946

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-nel 3050  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-po 5503  df-so 5504  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-1st 7831  df-2nd 7832  df-er 8498  df-en 8734  df-dom 8735  df-sdom 8736  df-pnf 11011  df-mnf 11012  df-xr 11013  df-ltxr 11014  df-le 11015  df-ioo 13083

This theorem is referenced by:  volioofmpt  43535  voliooicof  43537