fvvolioof - Mathbox for Glauco Siliprandi

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Theorem fvvolioof 45277

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 6715	. . 3 ⊢ (𝜑 → Fun 𝐹)
3		fvvolioof.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
4	1	fdmd 6722	. . . . 5 ⊢ (𝜑 → dom 𝐹 = 𝐴)
5	4	eqcomd 2732	. . . 4 ⊢ (𝜑 → 𝐴 = dom 𝐹)
6	3, 5	eleqtrd 2829	. . 3 ⊢ (𝜑 → 𝑋 ∈ dom 𝐹)
7		fvco 6983	. . 3 ⊢ ((Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹) → (((vol ∘ (,)) ∘ 𝐹)‘𝑋) = ((vol ∘ (,))‘(𝐹‘𝑋)))
8	2, 6, 7	syl2anc 583	. 2 ⊢ (𝜑 → (((vol ∘ (,)) ∘ 𝐹)‘𝑋) = ((vol ∘ (,))‘(𝐹‘𝑋)))
9		ioof 13430	. . . . 5 ⊢ (,):(ℝ^* × ℝ^*)⟶𝒫 ℝ
10		ffun 6714	. . . . 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 7081	. . . 4 ⊢ (𝜑 → (𝐹‘𝑋) ∈ (ℝ^* × ℝ^*))
14	9	fdmi 6723	. . . 4 ⊢ dom (,) = (ℝ^* × ℝ^*)
15	13, 14	eleqtrrdi 2838	. . 3 ⊢ (𝜑 → (𝐹‘𝑋) ∈ dom (,))
16		fvco 6983	. . 3 ⊢ ((Fun (,) ∧ (𝐹‘𝑋) ∈ dom (,)) → ((vol ∘ (,))‘(𝐹‘𝑋)) = (vol‘((,)‘(𝐹‘𝑋))))
17	12, 15, 16	syl2anc 583	. 2 ⊢ (𝜑 → ((vol ∘ (,))‘(𝐹‘𝑋)) = (vol‘((,)‘(𝐹‘𝑋))))
18		df-ov 7408	. . . . 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 2732	. . . . 5 ⊢ (𝜑 → ⟨(1^st ‘(𝐹‘𝑋)), (2^nd ‘(𝐹‘𝑋))⟩ = (𝐹‘𝑋))
23	22	fveq2d 6889	. . . 4 ⊢ (𝜑 → ((,)‘⟨(1^st ‘(𝐹‘𝑋)), (2^nd ‘(𝐹‘𝑋))⟩) = ((,)‘(𝐹‘𝑋)))
24	19, 23	eqtr2d 2767	. . 3 ⊢ (𝜑 → ((,)‘(𝐹‘𝑋)) = ((1^st ‘(𝐹‘𝑋))(,)(2^nd ‘(𝐹‘𝑋))))
25	24	fveq2d 6889	. 2 ⊢ (𝜑 → (vol‘((,)‘(𝐹‘𝑋))) = (vol‘((1^st ‘(𝐹‘𝑋))(,)(2^nd ‘(𝐹‘𝑋)))))
26	8, 17, 25	3eqtrd 2770	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 4597 ⟨cop 4629 × cxp 5667 dom cdm 5669 ∘ ccom 5673 Fun wfun 6531 ⟶wf 6533 ‘cfv 6537 (class class class)co 7405 1^st c1st 7972 2^nd c2nd 7973 ℝcr 11111 ℝ^*cxr 11251 (,)cioo 13330 volcvol 25347

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 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 ax-cnex 11168 ax-resscn 11169 ax-pre-lttri 11186 ax-pre-lttrn 11187

This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-nel 3041 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-po 5581 df-so 5582 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-ov 7408 df-oprab 7409 df-mpo 7410 df-1st 7974 df-2nd 7975 df-er 8705 df-en 8942 df-dom 8943 df-sdom 8944 df-pnf 11254 df-mnf 11255 df-xr 11256 df-ltxr 11257 df-le 11258 df-ioo 13334

This theorem is referenced by: volioofmpt 45282 voliooicof 45284

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