fvvolioof - Mathbox for Glauco Siliprandi

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Theorem fvvolioof 45424

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‘((1^st ‘(𝐹‘𝑋))(,)(2^nd ‘(𝐹‘𝑋)))))

**Proof of Theorem fvvolioof**
Step	Hyp	Ref	Expression
1		fvvolioof.f	. . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶(ℝ^* × ℝ^*))
2	1	ffund 6731	. . 3 ⊢ (𝜑 → Fun 𝐹)
3		fvvolioof.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
4	1	fdmd 6738	. . . . 5 ⊢ (𝜑 → dom 𝐹 = 𝐴)
5	4	eqcomd 2734	. . . 4 ⊢ (𝜑 → 𝐴 = dom 𝐹)
6	3, 5	eleqtrd 2831	. . 3 ⊢ (𝜑 → 𝑋 ∈ dom 𝐹)
7		fvco 7001	. . 3 ⊢ ((Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹) → (((vol ∘ (,)) ∘ 𝐹)‘𝑋) = ((vol ∘ (,))‘(𝐹‘𝑋)))
8	2, 6, 7	syl2anc 582	. 2 ⊢ (𝜑 → (((vol ∘ (,)) ∘ 𝐹)‘𝑋) = ((vol ∘ (,))‘(𝐹‘𝑋)))
9		ioof 13466	. . . . 5 ⊢ (,):(ℝ^* × ℝ^*)⟶𝒫 ℝ
10		ffun 6730	. . . . 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 7100	. . . 4 ⊢ (𝜑 → (𝐹‘𝑋) ∈ (ℝ^* × ℝ^*))
14	9	fdmi 6739	. . . 4 ⊢ dom (,) = (ℝ^* × ℝ^*)
15	13, 14	eleqtrrdi 2840	. . 3 ⊢ (𝜑 → (𝐹‘𝑋) ∈ dom (,))
16		fvco 7001	. . 3 ⊢ ((Fun (,) ∧ (𝐹‘𝑋) ∈ dom (,)) → ((vol ∘ (,))‘(𝐹‘𝑋)) = (vol‘((,)‘(𝐹‘𝑋))))
17	12, 15, 16	syl2anc 582	. 2 ⊢ (𝜑 → ((vol ∘ (,))‘(𝐹‘𝑋)) = (vol‘((,)‘(𝐹‘𝑋))))
18		df-ov 7429	. . . . 5 ⊢ ((1^st ‘(𝐹‘𝑋))(,)(2^nd ‘(𝐹‘𝑋))) = ((,)‘⟨(1^st ‘(𝐹‘𝑋)), (2^nd ‘(𝐹‘𝑋))⟩)
19	18	a1i 11	. . . 4 ⊢ (𝜑 → ((1^st ‘(𝐹‘𝑋))(,)(2^nd ‘(𝐹‘𝑋))) = ((,)‘⟨(1^st ‘(𝐹‘𝑋)), (2^nd ‘(𝐹‘𝑋))⟩))
20		1st2nd2 8040	. . . . . . 7 ⊢ ((𝐹‘𝑋) ∈ (ℝ^* × ℝ^*) → (𝐹‘𝑋) = ⟨(1^st ‘(𝐹‘𝑋)), (2^nd ‘(𝐹‘𝑋))⟩)
21	13, 20	syl 17	. . . . . 6 ⊢ (𝜑 → (𝐹‘𝑋) = ⟨(1^st ‘(𝐹‘𝑋)), (2^nd ‘(𝐹‘𝑋))⟩)
22	21	eqcomd 2734	. . . . 5 ⊢ (𝜑 → ⟨(1^st ‘(𝐹‘𝑋)), (2^nd ‘(𝐹‘𝑋))⟩ = (𝐹‘𝑋))
23	22	fveq2d 6906	. . . 4 ⊢ (𝜑 → ((,)‘⟨(1^st ‘(𝐹‘𝑋)), (2^nd ‘(𝐹‘𝑋))⟩) = ((,)‘(𝐹‘𝑋)))
24	19, 23	eqtr2d 2769	. . 3 ⊢ (𝜑 → ((,)‘(𝐹‘𝑋)) = ((1^st ‘(𝐹‘𝑋))(,)(2^nd ‘(𝐹‘𝑋))))
25	24	fveq2d 6906	. 2 ⊢ (𝜑 → (vol‘((,)‘(𝐹‘𝑋))) = (vol‘((1^st ‘(𝐹‘𝑋))(,)(2^nd ‘(𝐹‘𝑋)))))
26	8, 17, 25	3eqtrd 2772	1 ⊢ (𝜑 → (((vol ∘ (,)) ∘ 𝐹)‘𝑋) = (vol‘((1^st ‘(𝐹‘𝑋))(,)(2^nd ‘(𝐹‘𝑋)))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 𝒫 cpw 4606 ⟨cop 4638 × cxp 5680 dom cdm 5682 ∘ ccom 5686 Fun wfun 6547 ⟶wf 6549 ‘cfv 6553 (class class class)co 7426 1^st c1st 7999 2^nd c2nd 8000 ℝcr 11147 ℝ^*cxr 11287 (,)cioo 13366 volcvol 25420

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pow 5369 ax-pr 5433 ax-un 7748 ax-cnex 11204 ax-resscn 11205 ax-pre-lttri 11222 ax-pre-lttrn 11223

This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-pw 4608 df-sn 4633 df-pr 4635 df-op 4639 df-uni 4913 df-iun 5002 df-br 5153 df-opab 5215 df-mpt 5236 df-id 5580 df-po 5594 df-so 5595 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-ima 5695 df-iota 6505 df-fun 6555 df-fn 6556 df-f 6557 df-f1 6558 df-fo 6559 df-f1o 6560 df-fv 6561 df-ov 7429 df-oprab 7430 df-mpo 7431 df-1st 8001 df-2nd 8002 df-er 8733 df-en 8973 df-dom 8974 df-sdom 8975 df-pnf 11290 df-mnf 11291 df-xr 11292 df-ltxr 11293 df-le 11294 df-ioo 13370

This theorem is referenced by: volioofmpt 45429 voliooicof 45431

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