Theorem fvvolicof 46028

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 6655	. . 3 ⊢ (𝜑 → Fun 𝐹)
3		fvvolicof.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
4	1	fdmd 6661	. . . . 5 ⊢ (𝜑 → dom 𝐹 = 𝐴)
5	4	eqcomd 2737	. . . 4 ⊢ (𝜑 → 𝐴 = dom 𝐹)
6	3, 5	eleqtrd 2833	. . 3 ⊢ (𝜑 → 𝑋 ∈ dom 𝐹)
7		fvco 6920	. . 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 45255	. . . . 5 ⊢ [,):(ℝ* × ℝ*)⟶𝒫 ℝ*
10		ffun 6654	. . . . 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 7018	. . . 4 ⊢ (𝜑 → (𝐹‘𝑋) ∈ (ℝ* × ℝ*))
14	9	fdmi 6662	. . . 4 ⊢ dom [,) = (ℝ* × ℝ*)
15	13, 14	eleqtrrdi 2842	. . 3 ⊢ (𝜑 → (𝐹‘𝑋) ∈ dom [,))
16		fvco 6920	. . 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 7349	. . . . 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 7960	. . . . . . 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 2737	. . . . 5 ⊢ (𝜑 → ⟨(1st ‘(𝐹‘𝑋)), (2nd ‘(𝐹‘𝑋))⟩ = (𝐹‘𝑋))
23	22	fveq2d 6826	. . . 4 ⊢ (𝜑 → ([,)‘⟨(1st ‘(𝐹‘𝑋)), (2nd ‘(𝐹‘𝑋))⟩) = ([,)‘(𝐹‘𝑋)))
24	19, 23	eqtr2d 2767	. . 3 ⊢ (𝜑 → ([,)‘(𝐹‘𝑋)) = ((1st ‘(𝐹‘𝑋))[,)(2nd ‘(𝐹‘𝑋))))
25	24	fveq2d 6826	. 2 ⊢ (𝜑 → (vol‘([,)‘(𝐹‘𝑋))) = (vol‘((1st ‘(𝐹‘𝑋))[,)(2nd ‘(𝐹‘𝑋)))))
26	8, 17, 25	3eqtrd 2770	1 ⊢ (𝜑 → (((vol ∘ [,)) ∘ 𝐹)‘𝑋) = (vol‘((1st ‘(𝐹‘𝑋))[,)(2nd ‘(𝐹‘𝑋)))))

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2111  𝒫 cpw 4550  ⟨cop 4582   × cxp 5614  dom cdm 5616   ∘ ccom 5620  Fun wfun 6475  ⟶wf 6477  ‘cfv 6481  (class class class)co 7346  1^st c1st 7919  2^nd c2nd 7920  ℝ^*cxr 11142  [,)cico 13244  volcvol 25389

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pr 5370  ax-un 7668  ax-cnex 11059  ax-resscn 11060

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-pw 4552  df-sn 4577  df-pr 4579  df-op 4583  df-uni 4860  df-iun 4943  df-br 5092  df-opab 5154  df-mpt 5173  df-id 5511  df-xp 5622  df-rel 5623  df-cnv 5624  df-co 5625  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-fv 6489  df-ov 7349  df-oprab 7350  df-mpo 7351  df-1st 7921  df-2nd 7922  df-xr 11147  df-ico 13248

This theorem is referenced by:  voliooicof  46033  volicofmpt  46034