Theorem fvvolioof 45985

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 6715	. . 3 ⊢ (𝜑 → Fun 𝐹)
3		fvvolioof.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
4	1	fdmd 6721	. . . . 5 ⊢ (𝜑 → dom 𝐹 = 𝐴)
5	4	eqcomd 2742	. . . 4 ⊢ (𝜑 → 𝐴 = dom 𝐹)
6	3, 5	eleqtrd 2837	. . 3 ⊢ (𝜑 → 𝑋 ∈ dom 𝐹)
7		fvco 6982	. . 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 13469	. . . . 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 7080	. . . 4 ⊢ (𝜑 → (𝐹‘𝑋) ∈ (ℝ* × ℝ*))
14	9	fdmi 6722	. . . 4 ⊢ dom (,) = (ℝ* × ℝ*)
15	13, 14	eleqtrrdi 2846	. . 3 ⊢ (𝜑 → (𝐹‘𝑋) ∈ dom (,))
16		fvco 6982	. . 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 7413	. . . . 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 8032	. . . . . . 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 2742	. . . . 5 ⊢ (𝜑 → ⟨(1st ‘(𝐹‘𝑋)), (2nd ‘(𝐹‘𝑋))⟩ = (𝐹‘𝑋))
23	22	fveq2d 6885	. . . 4 ⊢ (𝜑 → ((,)‘⟨(1st ‘(𝐹‘𝑋)), (2nd ‘(𝐹‘𝑋))⟩) = ((,)‘(𝐹‘𝑋)))
24	19, 23	eqtr2d 2772	. . 3 ⊢ (𝜑 → ((,)‘(𝐹‘𝑋)) = ((1st ‘(𝐹‘𝑋))(,)(2nd ‘(𝐹‘𝑋))))
25	24	fveq2d 6885	. 2 ⊢ (𝜑 → (vol‘((,)‘(𝐹‘𝑋))) = (vol‘((1st ‘(𝐹‘𝑋))(,)(2nd ‘(𝐹‘𝑋)))))
26	8, 17, 25	3eqtrd 2775	1 ⊢ (𝜑 → (((vol ∘ (,)) ∘ 𝐹)‘𝑋) = (vol‘((1st ‘(𝐹‘𝑋))(,)(2nd ‘(𝐹‘𝑋)))))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  𝒫 cpw 4580  ⟨cop 4612   × cxp 5657  dom cdm 5659   ∘ ccom 5663  Fun wfun 6530  ⟶wf 6532  ‘cfv 6536  (class class class)co 7410  1^st c1st 7991  2^nd c2nd 7992  ℝcr 11133  ℝ^*cxr 11273  (,)cioo 13367  volcvol 25421

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pow 5340  ax-pr 5407  ax-un 7734  ax-cnex 11190  ax-resscn 11191  ax-pre-lttri 11208  ax-pre-lttrn 11209

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-sbc 3771  df-csb 3880  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-pw 4582  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-iun 4974  df-br 5125  df-opab 5187  df-mpt 5207  df-id 5553  df-po 5566  df-so 5567  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7413  df-oprab 7414  df-mpo 7415  df-1st 7993  df-2nd 7994  df-er 8724  df-en 8965  df-dom 8966  df-sdom 8967  df-pnf 11276  df-mnf 11277  df-xr 11278  df-ltxr 11279  df-le 11280  df-ioo 13371

This theorem is referenced by:  volioofmpt  45990  voliooicof  45992