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Theorem fvvolioof 38882
 Description: The function value of the Lebesgue measure of an open interval composed with a function. (Contributed by Glauco Siliprandi, 3-Mar-2021.)
Hypotheses
Ref Expression
fvvolioof.f (𝜑𝐹:𝐴⟶(ℝ* × ℝ*))
fvvolioof.x (𝜑𝑋𝐴)
Assertion
Ref Expression
fvvolioof (𝜑 → (((vol ∘ (,)) ∘ 𝐹)‘𝑋) = (vol‘((1st ‘(𝐹𝑋))(,)(2nd ‘(𝐹𝑋)))))

Proof of Theorem fvvolioof
StepHypRef Expression
1 fvvolioof.f . . . 4 (𝜑𝐹:𝐴⟶(ℝ* × ℝ*))
2 ffun 5961 . . . 4 (𝐹:𝐴⟶(ℝ* × ℝ*) → Fun 𝐹)
31, 2syl 17 . . 3 (𝜑 → Fun 𝐹)
4 fvvolioof.x . . . 4 (𝜑𝑋𝐴)
5 fdm 5964 . . . . . 6 (𝐹:𝐴⟶(ℝ* × ℝ*) → dom 𝐹 = 𝐴)
61, 5syl 17 . . . . 5 (𝜑 → dom 𝐹 = 𝐴)
76eqcomd 2616 . . . 4 (𝜑𝐴 = dom 𝐹)
84, 7eleqtrd 2690 . . 3 (𝜑𝑋 ∈ dom 𝐹)
9 fvco 6184 . . 3 ((Fun 𝐹𝑋 ∈ dom 𝐹) → (((vol ∘ (,)) ∘ 𝐹)‘𝑋) = ((vol ∘ (,))‘(𝐹𝑋)))
103, 8, 9syl2anc 691 . 2 (𝜑 → (((vol ∘ (,)) ∘ 𝐹)‘𝑋) = ((vol ∘ (,))‘(𝐹𝑋)))
11 ioof 12142 . . . . 5 (,):(ℝ* × ℝ*)⟶𝒫 ℝ
12 ffun 5961 . . . . 5 ((,):(ℝ* × ℝ*)⟶𝒫 ℝ → Fun (,))
1311, 12ax-mp 5 . . . 4 Fun (,)
1413a1i 11 . . 3 (𝜑 → Fun (,))
151, 4ffvelrnd 6268 . . . 4 (𝜑 → (𝐹𝑋) ∈ (ℝ* × ℝ*))
1611fdmi 5965 . . . 4 dom (,) = (ℝ* × ℝ*)
1715, 16syl6eleqr 2699 . . 3 (𝜑 → (𝐹𝑋) ∈ dom (,))
18 fvco 6184 . . 3 ((Fun (,) ∧ (𝐹𝑋) ∈ dom (,)) → ((vol ∘ (,))‘(𝐹𝑋)) = (vol‘((,)‘(𝐹𝑋))))
1914, 17, 18syl2anc 691 . 2 (𝜑 → ((vol ∘ (,))‘(𝐹𝑋)) = (vol‘((,)‘(𝐹𝑋))))
20 df-ov 6552 . . . . 5 ((1st ‘(𝐹𝑋))(,)(2nd ‘(𝐹𝑋))) = ((,)‘⟨(1st ‘(𝐹𝑋)), (2nd ‘(𝐹𝑋))⟩)
2120a1i 11 . . . 4 (𝜑 → ((1st ‘(𝐹𝑋))(,)(2nd ‘(𝐹𝑋))) = ((,)‘⟨(1st ‘(𝐹𝑋)), (2nd ‘(𝐹𝑋))⟩))
22 1st2nd2 7096 . . . . . . 7 ((𝐹𝑋) ∈ (ℝ* × ℝ*) → (𝐹𝑋) = ⟨(1st ‘(𝐹𝑋)), (2nd ‘(𝐹𝑋))⟩)
2315, 22syl 17 . . . . . 6 (𝜑 → (𝐹𝑋) = ⟨(1st ‘(𝐹𝑋)), (2nd ‘(𝐹𝑋))⟩)
2423eqcomd 2616 . . . . 5 (𝜑 → ⟨(1st ‘(𝐹𝑋)), (2nd ‘(𝐹𝑋))⟩ = (𝐹𝑋))
2524fveq2d 6107 . . . 4 (𝜑 → ((,)‘⟨(1st ‘(𝐹𝑋)), (2nd ‘(𝐹𝑋))⟩) = ((,)‘(𝐹𝑋)))
2621, 25eqtr2d 2645 . . 3 (𝜑 → ((,)‘(𝐹𝑋)) = ((1st ‘(𝐹𝑋))(,)(2nd ‘(𝐹𝑋))))
2726fveq2d 6107 . 2 (𝜑 → (vol‘((,)‘(𝐹𝑋))) = (vol‘((1st ‘(𝐹𝑋))(,)(2nd ‘(𝐹𝑋)))))
2810, 19, 273eqtrd 2648 1 (𝜑 → (((vol ∘ (,)) ∘ 𝐹)‘𝑋) = (vol‘((1st ‘(𝐹𝑋))(,)(2nd ‘(𝐹𝑋)))))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   = wceq 1475   ∈ wcel 1977  𝒫 cpw 4108  ⟨cop 4131   × cxp 5036  dom cdm 5038   ∘ ccom 5042  Fun wfun 5798  ⟶wf 5800  ‘cfv 5804  (class class class)co 6549  1st c1st 7057  2nd c2nd 7058  ℝcr 9814  ℝ*cxr 9952  (,)cioo 12046  volcvol 23039 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-cnex 9871  ax-resscn 9872  ax-pre-lttri 9889  ax-pre-lttrn 9890 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-nel 2783  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-po 4959  df-so 4960  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-er 7629  df-en 7842  df-dom 7843  df-sdom 7844  df-pnf 9955  df-mnf 9956  df-xr 9957  df-ltxr 9958  df-le 9959  df-ioo 12050 This theorem is referenced by:  volioofmpt  38887  voliooicof  38889
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