Theorem fvvolicof 45912

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 6751	. . 3 ⊢ (𝜑 → Fun 𝐹)
3		fvvolicof.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
4	1	fdmd 6757	. . . . 5 ⊢ (𝜑 → dom 𝐹 = 𝐴)
5	4	eqcomd 2746	. . . 4 ⊢ (𝜑 → 𝐴 = dom 𝐹)
6	3, 5	eleqtrd 2846	. . 3 ⊢ (𝜑 → 𝑋 ∈ dom 𝐹)
7		fvco 7020	. . 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		icof 45126	. . . . 5 ⊢ [,):(ℝ* × ℝ*)⟶𝒫 ℝ*
10		ffun 6750	. . . . 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 7119	. . . 4 ⊢ (𝜑 → (𝐹‘𝑋) ∈ (ℝ* × ℝ*))
14	9	fdmi 6758	. . . 4 ⊢ dom [,) = (ℝ* × ℝ*)
15	13, 14	eleqtrrdi 2855	. . 3 ⊢ (𝜑 → (𝐹‘𝑋) ∈ dom [,))
16		fvco 7020	. . 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 7451	. . . . 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 8069	. . . . . . 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 2746	. . . . 5 ⊢ (𝜑 → ⟨(1st ‘(𝐹‘𝑋)), (2nd ‘(𝐹‘𝑋))⟩ = (𝐹‘𝑋))
23	22	fveq2d 6924	. . . 4 ⊢ (𝜑 → ([,)‘⟨(1st ‘(𝐹‘𝑋)), (2nd ‘(𝐹‘𝑋))⟩) = ([,)‘(𝐹‘𝑋)))
24	19, 23	eqtr2d 2781	. . 3 ⊢ (𝜑 → ([,)‘(𝐹‘𝑋)) = ((1st ‘(𝐹‘𝑋))[,)(2nd ‘(𝐹‘𝑋))))
25	24	fveq2d 6924	. 2 ⊢ (𝜑 → (vol‘([,)‘(𝐹‘𝑋))) = (vol‘((1st ‘(𝐹‘𝑋))[,)(2nd ‘(𝐹‘𝑋)))))
26	8, 17, 25	3eqtrd 2784	1 ⊢ (𝜑 → (((vol ∘ [,)) ∘ 𝐹)‘𝑋) = (vol‘((1st ‘(𝐹‘𝑋))[,)(2nd ‘(𝐹‘𝑋)))))

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2108  𝒫 cpw 4622  ⟨cop 4654   × cxp 5698  dom cdm 5700   ∘ ccom 5704  Fun wfun 6567  ⟶wf 6569  ‘cfv 6573  (class class class)co 7448  1^st c1st 8028  2^nd c2nd 8029  ℝ^*cxr 11323  [,)cico 13409  volcvol 25517

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7770  ax-cnex 11240  ax-resscn 11241

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-1st 8030  df-2nd 8031  df-xr 11328  df-ico 13413

This theorem is referenced by:  voliooicof  45917  volicofmpt  45918