Theorem fvvolicof 45989

Description: The function value of the Lebesgue measure of a left-closed right-open interval composed with a function. (Contributed by Glauco Siliprandi, 3-Mar-2021.)

Hypotheses

Ref	Expression
fvvolicof.f	⊢ (𝜑 → 𝐹:𝐴⟶(ℝ* × ℝ*))
fvvolicof.x	⊢ (𝜑 → 𝑋 ∈ 𝐴)

Assertion

Ref	Expression
fvvolicof	⊢ (𝜑 → (((vol ∘ [,)) ∘ 𝐹)‘𝑋) = (vol‘((1st ‘(𝐹‘𝑋))[,)(2nd ‘(𝐹‘𝑋)))))

Proof of Theorem fvvolicof

Step	Hyp	Ref	Expression
1		fvvolicof.f	. . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶(ℝ* × ℝ*))
2	1	ffund 6692	. . 3 ⊢ (𝜑 → Fun 𝐹)
3		fvvolicof.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
4	1	fdmd 6698	. . . . 5 ⊢ (𝜑 → dom 𝐹 = 𝐴)
5	4	eqcomd 2735	. . . 4 ⊢ (𝜑 → 𝐴 = dom 𝐹)
6	3, 5	eleqtrd 2830	. . 3 ⊢ (𝜑 → 𝑋 ∈ dom 𝐹)
7		fvco 6959	. . 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		icof 45213	. . . . 5 ⊢ [,):(ℝ* × ℝ*)⟶𝒫 ℝ*
10		ffun 6691	. . . . 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 7057	. . . 4 ⊢ (𝜑 → (𝐹‘𝑋) ∈ (ℝ* × ℝ*))
14	9	fdmi 6699	. . . 4 ⊢ dom [,) = (ℝ* × ℝ*)
15	13, 14	eleqtrrdi 2839	. . 3 ⊢ (𝜑 → (𝐹‘𝑋) ∈ dom [,))
16		fvco 6959	. . 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 7390	. . . . 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 8007	. . . . . . 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 2735	. . . . 5 ⊢ (𝜑 → ⟨(1st ‘(𝐹‘𝑋)), (2nd ‘(𝐹‘𝑋))⟩ = (𝐹‘𝑋))
23	22	fveq2d 6862	. . . 4 ⊢ (𝜑 → ([,)‘⟨(1st ‘(𝐹‘𝑋)), (2nd ‘(𝐹‘𝑋))⟩) = ([,)‘(𝐹‘𝑋)))
24	19, 23	eqtr2d 2765	. . 3 ⊢ (𝜑 → ([,)‘(𝐹‘𝑋)) = ((1st ‘(𝐹‘𝑋))[,)(2nd ‘(𝐹‘𝑋))))
25	24	fveq2d 6862	. 2 ⊢ (𝜑 → (vol‘([,)‘(𝐹‘𝑋))) = (vol‘((1st ‘(𝐹‘𝑋))[,)(2nd ‘(𝐹‘𝑋)))))
26	8, 17, 25	3eqtrd 2768	1 ⊢ (𝜑 → (((vol ∘ [,)) ∘ 𝐹)‘𝑋) = (vol‘((1st ‘(𝐹‘𝑋))[,)(2nd ‘(𝐹‘𝑋)))))

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  𝒫 cpw 4563  ⟨cop 4595   × cxp 5636  dom cdm 5638   ∘ ccom 5642  Fun wfun 6505  ⟶wf 6507  ‘cfv 6511  (class class class)co 7387  1^st c1st 7966  2^nd c2nd 7967  ℝ^*cxr 11207  [,)cico 13308  volcvol 25364

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 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387  ax-un 7711  ax-cnex 11124  ax-resscn 11125

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-fv 6519  df-ov 7390  df-oprab 7391  df-mpo 7392  df-1st 7968  df-2nd 7969  df-xr 11212  df-ico 13312

This theorem is referenced by:  voliooicof  45994  volicofmpt  45995