Theorem fvvolicof 40208

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		ffun 6048	. . . 4 ⊢ (𝐹:𝐴⟶(ℝ* × ℝ*) → Fun 𝐹)
3	1, 2	syl 17	. . 3 ⊢ (𝜑 → Fun 𝐹)
4		fvvolicof.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
5		fdm 6051	. . . . . 6 ⊢ (𝐹:𝐴⟶(ℝ* × ℝ*) → dom 𝐹 = 𝐴)
6	1, 5	syl 17	. . . . 5 ⊢ (𝜑 → dom 𝐹 = 𝐴)
7	6	eqcomd 2628	. . . 4 ⊢ (𝜑 → 𝐴 = dom 𝐹)
8	4, 7	eleqtrd 2703	. . 3 ⊢ (𝜑 → 𝑋 ∈ dom 𝐹)
9		fvco 6274	. . 3 ⊢ ((Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹) → (((vol ∘ [,)) ∘ 𝐹)‘𝑋) = ((vol ∘ [,))‘(𝐹‘𝑋)))
10	3, 8, 9	syl2anc 693	. 2 ⊢ (𝜑 → (((vol ∘ [,)) ∘ 𝐹)‘𝑋) = ((vol ∘ [,))‘(𝐹‘𝑋)))
11		icof 39411	. . . . 5 ⊢ [,):(ℝ* × ℝ*)⟶𝒫 ℝ*
12		ffun 6048	. . . . 5 ⊢ ([,):(ℝ* × ℝ*)⟶𝒫 ℝ* → Fun [,))
13	11, 12	ax-mp 5	. . . 4 ⊢ Fun [,)
14	13	a1i 11	. . 3 ⊢ (𝜑 → Fun [,))
15	1, 4	ffvelrnd 6360	. . . 4 ⊢ (𝜑 → (𝐹‘𝑋) ∈ (ℝ* × ℝ*))
16	11	fdmi 6052	. . . 4 ⊢ dom [,) = (ℝ* × ℝ*)
17	15, 16	syl6eleqr 2712	. . 3 ⊢ (𝜑 → (𝐹‘𝑋) ∈ dom [,))
18		fvco 6274	. . 3 ⊢ ((Fun [,) ∧ (𝐹‘𝑋) ∈ dom [,)) → ((vol ∘ [,))‘(𝐹‘𝑋)) = (vol‘([,)‘(𝐹‘𝑋))))
19	14, 17, 18	syl2anc 693	. 2 ⊢ (𝜑 → ((vol ∘ [,))‘(𝐹‘𝑋)) = (vol‘([,)‘(𝐹‘𝑋))))
20		df-ov 6653	. . . . 5 ⊢ ((1st ‘(𝐹‘𝑋))[,)(2nd ‘(𝐹‘𝑋))) = ([,)‘⟨(1st ‘(𝐹‘𝑋)), (2nd ‘(𝐹‘𝑋))⟩)
21	20	a1i 11	. . . 4 ⊢ (𝜑 → ((1st ‘(𝐹‘𝑋))[,)(2nd ‘(𝐹‘𝑋))) = ([,)‘⟨(1st ‘(𝐹‘𝑋)), (2nd ‘(𝐹‘𝑋))⟩))
22		1st2nd2 7205	. . . . . . 7 ⊢ ((𝐹‘𝑋) ∈ (ℝ* × ℝ*) → (𝐹‘𝑋) = ⟨(1st ‘(𝐹‘𝑋)), (2nd ‘(𝐹‘𝑋))⟩)
23	15, 22	syl 17	. . . . . 6 ⊢ (𝜑 → (𝐹‘𝑋) = ⟨(1st ‘(𝐹‘𝑋)), (2nd ‘(𝐹‘𝑋))⟩)
24	23	eqcomd 2628	. . . . 5 ⊢ (𝜑 → ⟨(1st ‘(𝐹‘𝑋)), (2nd ‘(𝐹‘𝑋))⟩ = (𝐹‘𝑋))
25	24	fveq2d 6195	. . . 4 ⊢ (𝜑 → ([,)‘⟨(1st ‘(𝐹‘𝑋)), (2nd ‘(𝐹‘𝑋))⟩) = ([,)‘(𝐹‘𝑋)))
26	21, 25	eqtr2d 2657	. . 3 ⊢ (𝜑 → ([,)‘(𝐹‘𝑋)) = ((1st ‘(𝐹‘𝑋))[,)(2nd ‘(𝐹‘𝑋))))
27	26	fveq2d 6195	. 2 ⊢ (𝜑 → (vol‘([,)‘(𝐹‘𝑋))) = (vol‘((1st ‘(𝐹‘𝑋))[,)(2nd ‘(𝐹‘𝑋)))))
28	10, 19, 27	3eqtrd 2660	1 ⊢ (𝜑 → (((vol ∘ [,)) ∘ 𝐹)‘𝑋) = (vol‘((1st ‘(𝐹‘𝑋))[,)(2nd ‘(𝐹‘𝑋)))))

Syntax hints:   → wi 4   = wceq 1483   ∈ wcel 1990  𝒫 cpw 4158  ⟨cop 4183   × cxp 5112  dom cdm 5114   ∘ ccom 5118  Fun wfun 5882  ⟶wf 5884  ‘cfv 5888  (class class class)co 6650  1^st c1st 7166  2^nd c2nd 7167  ℝ^*cxr 10073  [,)cico 12177  volcvol 23232

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-xr 10078  df-ico 12181

This theorem is referenced by:  voliooicof  40213  volicofmpt  40214