Theorem fvvolicof 47094

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 6714	. . 3 ⊢ (𝜑 → Fun 𝐹)
3		fvvolicof.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
4	1	fdmd 6720	. . . . 5 ⊢ (𝜑 → dom 𝐹 = 𝐴)
5	4	eqcomd 2766	. . . 4 ⊢ (𝜑 → 𝐴 = dom 𝐹)
6	3, 5	eleqtrd 2862	. . 3 ⊢ (𝜑 → 𝑋 ∈ dom 𝐹)
7		fvco 6983	. . 3 ⊢ ((Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹) → (((vol ∘ [,)) ∘ 𝐹)‘𝑋) = ((vol ∘ [,))‘(𝐹‘𝑋)))
8	2, 6, 7	syl2anc 596	. 2 ⊢ (𝜑 → (((vol ∘ [,)) ∘ 𝐹)‘𝑋) = ((vol ∘ [,))‘(𝐹‘𝑋)))
9		icof 46325	. . . . 5 ⊢ [,):(ℝ* × ℝ*)⟶𝒫 ℝ*
10		ffun 6712	. . . . 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 7085	. . . 4 ⊢ (𝜑 → (𝐹‘𝑋) ∈ (ℝ* × ℝ*))
14	9	fdmi 6721	. . . 4 ⊢ dom [,) = (ℝ* × ℝ*)
15	13, 14	eleqtrrdi 2871	. . 3 ⊢ (𝜑 → (𝐹‘𝑋) ∈ dom [,))
16		fvco 6983	. . 3 ⊢ ((Fun [,) ∧ (𝐹‘𝑋) ∈ dom [,)) → ((vol ∘ [,))‘(𝐹‘𝑋)) = (vol‘([,)‘(𝐹‘𝑋))))
17	12, 15, 16	syl2anc 596	. 2 ⊢ (𝜑 → ((vol ∘ [,))‘(𝐹‘𝑋)) = (vol‘([,)‘(𝐹‘𝑋))))
18		df-ov 7423	. . . . 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 8040	. . . . . . 7 ⊢ ((𝐹‘𝑋) ∈ (ℝ* × ℝ*) → (𝐹‘𝑋) = ⟨(1st ‘(𝐹‘𝑋)), (2nd ‘(𝐹‘𝑋))⟩)
21	13, 20	syl 18	. . . . . 6 ⊢ (𝜑 → (𝐹‘𝑋) = ⟨(1st ‘(𝐹‘𝑋)), (2nd ‘(𝐹‘𝑋))⟩)
22	21	eqcomd 2766	. . . . 5 ⊢ (𝜑 → ⟨(1st ‘(𝐹‘𝑋)), (2nd ‘(𝐹‘𝑋))⟩ = (𝐹‘𝑋))
23	22	fveq2d 6889	. . . 4 ⊢ (𝜑 → ([,)‘⟨(1st ‘(𝐹‘𝑋)), (2nd ‘(𝐹‘𝑋))⟩) = ([,)‘(𝐹‘𝑋)))
24	19, 23	eqtr2d 2796	. . 3 ⊢ (𝜑 → ([,)‘(𝐹‘𝑋)) = ((1st ‘(𝐹‘𝑋))[,)(2nd ‘(𝐹‘𝑋))))
25	24	fveq2d 6889	. 2 ⊢ (𝜑 → (vol‘([,)‘(𝐹‘𝑋))) = (vol‘((1st ‘(𝐹‘𝑋))[,)(2nd ‘(𝐹‘𝑋)))))
26	8, 17, 25	3eqtrd 2799	1 ⊢ (𝜑 → (((vol ∘ [,)) ∘ 𝐹)‘𝑋) = (vol‘((1st ‘(𝐹‘𝑋))[,)(2nd ‘(𝐹‘𝑋)))))

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∈ wcel 2144  𝒫 cpw 4556  ⟨cop 4834   × cxp 5648  dom cdm 5650   ∘ ccom 5654  Fun wfun 6532  ⟶wf 6534  ‘cfv 6538  (class class class)co 7420  1^st c1st 7999  2^nd c2nd 8000  ℝ^*cxr 11345  [,)cico 13481  volcvol 25789

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390  ax-un 7751  ax-cnex 11259  ax-resscn 11260

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-fv 6546  df-ov 7423  df-oprab 7424  df-mpo 7425  df-1st 8001  df-2nd 8002  df-xr 11350  df-ico 13485

This theorem is used by:  voliooicof  47099  volicofmpt  47100