Theorem fvvolioof 46432

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 6659	. . 3 ⊢ (𝜑 → Fun 𝐹)
3		fvvolioof.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
4	1	fdmd 6665	. . . . 5 ⊢ (𝜑 → dom 𝐹 = 𝐴)
5	4	eqcomd 2745	. . . 4 ⊢ (𝜑 → 𝐴 = dom 𝐹)
6	3, 5	eleqtrd 2841	. . 3 ⊢ (𝜑 → 𝑋 ∈ dom 𝐹)
7		fvco 6925	. . 3 ⊢ ((Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹) → (((vol ∘ (,)) ∘ 𝐹)‘𝑋) = ((vol ∘ (,))‘(𝐹‘𝑋)))
8	2, 6, 7	syl2anc 590	. 2 ⊢ (𝜑 → (((vol ∘ (,)) ∘ 𝐹)‘𝑋) = ((vol ∘ (,))‘(𝐹‘𝑋)))
9		ioof 13391	. . . . 5 ⊢ (,):(ℝ* × ℝ*)⟶𝒫 ℝ
10		ffun 6658	. . . . 5 ⊢ ((,):(ℝ* × ℝ*)⟶𝒫 ℝ → Fun (,))
11	9, 10	ax-mp 5	. . . 4 ⊢ Fun (,)
12	11	a1i 11	. . 3 ⊢ (𝜑 → Fun (,))
13	1, 3	ffvelcdmd 7026	. . . 4 ⊢ (𝜑 → (𝐹‘𝑋) ∈ (ℝ* × ℝ*))
14	9	fdmi 6666	. . . 4 ⊢ dom (,) = (ℝ* × ℝ*)
15	13, 14	eleqtrrdi 2850	. . 3 ⊢ (𝜑 → (𝐹‘𝑋) ∈ dom (,))
16		fvco 6925	. . 3 ⊢ ((Fun (,) ∧ (𝐹‘𝑋) ∈ dom (,)) → ((vol ∘ (,))‘(𝐹‘𝑋)) = (vol‘((,)‘(𝐹‘𝑋))))
17	12, 15, 16	syl2anc 590	. 2 ⊢ (𝜑 → ((vol ∘ (,))‘(𝐹‘𝑋)) = (vol‘((,)‘(𝐹‘𝑋))))
18		df-ov 7359	. . . . 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 7970	. . . . . . 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 2745	. . . . 5 ⊢ (𝜑 → ⟨(1st ‘(𝐹‘𝑋)), (2nd ‘(𝐹‘𝑋))⟩ = (𝐹‘𝑋))
23	22	fveq2d 6831	. . . 4 ⊢ (𝜑 → ((,)‘⟨(1st ‘(𝐹‘𝑋)), (2nd ‘(𝐹‘𝑋))⟩) = ((,)‘(𝐹‘𝑋)))
24	19, 23	eqtr2d 2775	. . 3 ⊢ (𝜑 → ((,)‘(𝐹‘𝑋)) = ((1st ‘(𝐹‘𝑋))(,)(2nd ‘(𝐹‘𝑋))))
25	24	fveq2d 6831	. 2 ⊢ (𝜑 → (vol‘((,)‘(𝐹‘𝑋))) = (vol‘((1st ‘(𝐹‘𝑋))(,)(2nd ‘(𝐹‘𝑋)))))
26	8, 17, 25	3eqtrd 2778	1 ⊢ (𝜑 → (((vol ∘ (,)) ∘ 𝐹)‘𝑋) = (vol‘((1st ‘(𝐹‘𝑋))(,)(2nd ‘(𝐹‘𝑋)))))

Syntax hints:   → wi 4   = wceq 1547   ∈ wcel 2119  𝒫 cpw 4529  ⟨cop 4561   × cxp 5616  dom cdm 5618   ∘ ccom 5622  Fun wfun 6479  ⟶wf 6481  ‘cfv 6485  (class class class)co 7356  1^st c1st 7929  2^nd c2nd 7930  ℝcr 11028  ℝ^*cxr 11169  (,)cioo 13289  volcvol 25448

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678  ax-cnex 11085  ax-resscn 11086  ax-pre-lttri 11103  ax-pre-lttrn 11104

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-nel 3039  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-po 5526  df-so 5527  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-ov 7359  df-oprab 7360  df-mpo 7361  df-1st 7931  df-2nd 7932  df-er 8633  df-en 8884  df-dom 8885  df-sdom 8886  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175  df-le 11176  df-ioo 13293

This theorem is referenced by:  volioofmpt  46437  voliooicof  46439