fvvolioof - Mathbox for Glauco Siliprandi

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Theorem fvvolioof 38882

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		ffun 5961	. . . 4 ⊢ (𝐹:𝐴⟶(ℝ^* × ℝ^*) → Fun 𝐹)
3	1, 2	syl 17	. . 3 ⊢ (𝜑 → Fun 𝐹)
4		fvvolioof.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
5		fdm 5964	. . . . . 6 ⊢ (𝐹:𝐴⟶(ℝ^* × ℝ^*) → dom 𝐹 = 𝐴)
6	1, 5	syl 17	. . . . 5 ⊢ (𝜑 → dom 𝐹 = 𝐴)
7	6	eqcomd 2616	. . . 4 ⊢ (𝜑 → 𝐴 = dom 𝐹)
8	4, 7	eleqtrd 2690	. . 3 ⊢ (𝜑 → 𝑋 ∈ dom 𝐹)
9		fvco 6184	. . 3 ⊢ ((Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹) → (((vol ∘ (,)) ∘ 𝐹)‘𝑋) = ((vol ∘ (,))‘(𝐹‘𝑋)))
10	3, 8, 9	syl2anc 691	. 2 ⊢ (𝜑 → (((vol ∘ (,)) ∘ 𝐹)‘𝑋) = ((vol ∘ (,))‘(𝐹‘𝑋)))
11		ioof 12142	. . . . 5 ⊢ (,):(ℝ^* × ℝ^*)⟶𝒫 ℝ
12		ffun 5961	. . . . 5 ⊢ ((,):(ℝ^* × ℝ^*)⟶𝒫 ℝ → Fun (,))
13	11, 12	ax-mp 5	. . . 4 ⊢ Fun (,)
14	13	a1i 11	. . 3 ⊢ (𝜑 → Fun (,))
15	1, 4	ffvelrnd 6268	. . . 4 ⊢ (𝜑 → (𝐹‘𝑋) ∈ (ℝ^* × ℝ^*))
16	11	fdmi 5965	. . . 4 ⊢ dom (,) = (ℝ^* × ℝ^*)
17	15, 16	syl6eleqr 2699	. . 3 ⊢ (𝜑 → (𝐹‘𝑋) ∈ dom (,))
18		fvco 6184	. . 3 ⊢ ((Fun (,) ∧ (𝐹‘𝑋) ∈ dom (,)) → ((vol ∘ (,))‘(𝐹‘𝑋)) = (vol‘((,)‘(𝐹‘𝑋))))
19	14, 17, 18	syl2anc 691	. 2 ⊢ (𝜑 → ((vol ∘ (,))‘(𝐹‘𝑋)) = (vol‘((,)‘(𝐹‘𝑋))))
20		df-ov 6552	. . . . 5 ⊢ ((1^st ‘(𝐹‘𝑋))(,)(2^nd ‘(𝐹‘𝑋))) = ((,)‘⟨(1^st ‘(𝐹‘𝑋)), (2^nd ‘(𝐹‘𝑋))⟩)
21	20	a1i 11	. . . 4 ⊢ (𝜑 → ((1^st ‘(𝐹‘𝑋))(,)(2^nd ‘(𝐹‘𝑋))) = ((,)‘⟨(1^st ‘(𝐹‘𝑋)), (2^nd ‘(𝐹‘𝑋))⟩))
22		1st2nd2 7096	. . . . . . 7 ⊢ ((𝐹‘𝑋) ∈ (ℝ^* × ℝ^*) → (𝐹‘𝑋) = ⟨(1^st ‘(𝐹‘𝑋)), (2^nd ‘(𝐹‘𝑋))⟩)
23	15, 22	syl 17	. . . . . 6 ⊢ (𝜑 → (𝐹‘𝑋) = ⟨(1^st ‘(𝐹‘𝑋)), (2^nd ‘(𝐹‘𝑋))⟩)
24	23	eqcomd 2616	. . . . 5 ⊢ (𝜑 → ⟨(1^st ‘(𝐹‘𝑋)), (2^nd ‘(𝐹‘𝑋))⟩ = (𝐹‘𝑋))
25	24	fveq2d 6107	. . . 4 ⊢ (𝜑 → ((,)‘⟨(1^st ‘(𝐹‘𝑋)), (2^nd ‘(𝐹‘𝑋))⟩) = ((,)‘(𝐹‘𝑋)))
26	21, 25	eqtr2d 2645	. . 3 ⊢ (𝜑 → ((,)‘(𝐹‘𝑋)) = ((1^st ‘(𝐹‘𝑋))(,)(2^nd ‘(𝐹‘𝑋))))
27	26	fveq2d 6107	. 2 ⊢ (𝜑 → (vol‘((,)‘(𝐹‘𝑋))) = (vol‘((1^st ‘(𝐹‘𝑋))(,)(2^nd ‘(𝐹‘𝑋)))))
28	10, 19, 27	3eqtrd 2648	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 1475 ∈ wcel 1977 𝒫 cpw 4108 ⟨cop 4131 × cxp 5036 dom cdm 5038 ∘ ccom 5042 Fun wfun 5798 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 1^st c1st 7057 2^nd c2nd 7058 ℝcr 9814 ℝ^*cxr 9952 (,)cioo 12046 volcvol 23039

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-8 1979 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-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-pre-lttri 9889 ax-pre-lttrn 9890

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 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-ne 2782 df-nel 2783 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-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-po 4959 df-so 4960 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-1st 7059 df-2nd 7060 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-ioo 12050

This theorem is referenced by: volioofmpt 38887 voliooicof 38889

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