fvvolioof - Mathbox for Glauco Siliprandi

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Theorem fvvolioof 44695

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 6721	. . 3 ⊢ (𝜑 → Fun 𝐹)
3		fvvolioof.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
4	1	fdmd 6728	. . . . 5 ⊢ (𝜑 → dom 𝐹 = 𝐴)
5	4	eqcomd 2738	. . . 4 ⊢ (𝜑 → 𝐴 = dom 𝐹)
6	3, 5	eleqtrd 2835	. . 3 ⊢ (𝜑 → 𝑋 ∈ dom 𝐹)
7		fvco 6989	. . 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 13423	. . . . 5 ⊢ (,):(ℝ^* × ℝ^*)⟶𝒫 ℝ
10		ffun 6720	. . . . 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 7087	. . . 4 ⊢ (𝜑 → (𝐹‘𝑋) ∈ (ℝ^* × ℝ^*))
14	9	fdmi 6729	. . . 4 ⊢ dom (,) = (ℝ^* × ℝ^*)
15	13, 14	eleqtrrdi 2844	. . 3 ⊢ (𝜑 → (𝐹‘𝑋) ∈ dom (,))
16		fvco 6989	. . 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 7411	. . . . 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 8013	. . . . . . 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 2738	. . . . 5 ⊢ (𝜑 → ⟨(1^st ‘(𝐹‘𝑋)), (2^nd ‘(𝐹‘𝑋))⟩ = (𝐹‘𝑋))
23	22	fveq2d 6895	. . . 4 ⊢ (𝜑 → ((,)‘⟨(1^st ‘(𝐹‘𝑋)), (2^nd ‘(𝐹‘𝑋))⟩) = ((,)‘(𝐹‘𝑋)))
24	19, 23	eqtr2d 2773	. . 3 ⊢ (𝜑 → ((,)‘(𝐹‘𝑋)) = ((1^st ‘(𝐹‘𝑋))(,)(2^nd ‘(𝐹‘𝑋))))
25	24	fveq2d 6895	. 2 ⊢ (𝜑 → (vol‘((,)‘(𝐹‘𝑋))) = (vol‘((1^st ‘(𝐹‘𝑋))(,)(2^nd ‘(𝐹‘𝑋)))))
26	8, 17, 25	3eqtrd 2776	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 1541 ∈ wcel 2106 𝒫 cpw 4602 ⟨cop 4634 × cxp 5674 dom cdm 5676 ∘ ccom 5680 Fun wfun 6537 ⟶wf 6539 ‘cfv 6543 (class class class)co 7408 1^st c1st 7972 2^nd c2nd 7973 ℝcr 11108 ℝ^*cxr 11246 (,)cioo 13323 volcvol 24979

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 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 2703 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7724 ax-cnex 11165 ax-resscn 11166 ax-pre-lttri 11183 ax-pre-lttrn 11184

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-iun 4999 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-po 5588 df-so 5589 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7411 df-oprab 7412 df-mpo 7413 df-1st 7974 df-2nd 7975 df-er 8702 df-en 8939 df-dom 8940 df-sdom 8941 df-pnf 11249 df-mnf 11250 df-xr 11251 df-ltxr 11252 df-le 11253 df-ioo 13327

This theorem is referenced by: volioofmpt 44700 voliooicof 44702

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