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Theorem fvvolioof 45945
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*))
21ffund 6741 . . 3 (𝜑 → Fun 𝐹)
3 fvvolioof.x . . . 4 (𝜑𝑋𝐴)
41fdmd 6747 . . . . 5 (𝜑 → dom 𝐹 = 𝐴)
54eqcomd 2741 . . . 4 (𝜑𝐴 = dom 𝐹)
63, 5eleqtrd 2841 . . 3 (𝜑𝑋 ∈ dom 𝐹)
7 fvco 7007 . . 3 ((Fun 𝐹𝑋 ∈ dom 𝐹) → (((vol ∘ (,)) ∘ 𝐹)‘𝑋) = ((vol ∘ (,))‘(𝐹𝑋)))
82, 6, 7syl2anc 584 . 2 (𝜑 → (((vol ∘ (,)) ∘ 𝐹)‘𝑋) = ((vol ∘ (,))‘(𝐹𝑋)))
9 ioof 13484 . . . . 5 (,):(ℝ* × ℝ*)⟶𝒫 ℝ
10 ffun 6740 . . . . 5 ((,):(ℝ* × ℝ*)⟶𝒫 ℝ → Fun (,))
119, 10ax-mp 5 . . . 4 Fun (,)
1211a1i 11 . . 3 (𝜑 → Fun (,))
131, 3ffvelcdmd 7105 . . . 4 (𝜑 → (𝐹𝑋) ∈ (ℝ* × ℝ*))
149fdmi 6748 . . . 4 dom (,) = (ℝ* × ℝ*)
1513, 14eleqtrrdi 2850 . . 3 (𝜑 → (𝐹𝑋) ∈ dom (,))
16 fvco 7007 . . 3 ((Fun (,) ∧ (𝐹𝑋) ∈ dom (,)) → ((vol ∘ (,))‘(𝐹𝑋)) = (vol‘((,)‘(𝐹𝑋))))
1712, 15, 16syl2anc 584 . 2 (𝜑 → ((vol ∘ (,))‘(𝐹𝑋)) = (vol‘((,)‘(𝐹𝑋))))
18 df-ov 7434 . . . . 5 ((1st ‘(𝐹𝑋))(,)(2nd ‘(𝐹𝑋))) = ((,)‘⟨(1st ‘(𝐹𝑋)), (2nd ‘(𝐹𝑋))⟩)
1918a1i 11 . . . 4 (𝜑 → ((1st ‘(𝐹𝑋))(,)(2nd ‘(𝐹𝑋))) = ((,)‘⟨(1st ‘(𝐹𝑋)), (2nd ‘(𝐹𝑋))⟩))
20 1st2nd2 8052 . . . . . . 7 ((𝐹𝑋) ∈ (ℝ* × ℝ*) → (𝐹𝑋) = ⟨(1st ‘(𝐹𝑋)), (2nd ‘(𝐹𝑋))⟩)
2113, 20syl 17 . . . . . 6 (𝜑 → (𝐹𝑋) = ⟨(1st ‘(𝐹𝑋)), (2nd ‘(𝐹𝑋))⟩)
2221eqcomd 2741 . . . . 5 (𝜑 → ⟨(1st ‘(𝐹𝑋)), (2nd ‘(𝐹𝑋))⟩ = (𝐹𝑋))
2322fveq2d 6911 . . . 4 (𝜑 → ((,)‘⟨(1st ‘(𝐹𝑋)), (2nd ‘(𝐹𝑋))⟩) = ((,)‘(𝐹𝑋)))
2419, 23eqtr2d 2776 . . 3 (𝜑 → ((,)‘(𝐹𝑋)) = ((1st ‘(𝐹𝑋))(,)(2nd ‘(𝐹𝑋))))
2524fveq2d 6911 . 2 (𝜑 → (vol‘((,)‘(𝐹𝑋))) = (vol‘((1st ‘(𝐹𝑋))(,)(2nd ‘(𝐹𝑋)))))
268, 17, 253eqtrd 2779 1 (𝜑 → (((vol ∘ (,)) ∘ 𝐹)‘𝑋) = (vol‘((1st ‘(𝐹𝑋))(,)(2nd ‘(𝐹𝑋)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  𝒫 cpw 4605  cop 4637   × cxp 5687  dom cdm 5689  ccom 5693  Fun wfun 6557  wf 6559  cfv 6563  (class class class)co 7431  1st c1st 8011  2nd c2nd 8012  cr 11152  *cxr 11292  (,)cioo 13384  volcvol 25512
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-cnex 11209  ax-resscn 11210  ax-pre-lttri 11227  ax-pre-lttrn 11228
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-po 5597  df-so 5598  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-er 8744  df-en 8985  df-dom 8986  df-sdom 8987  df-pnf 11295  df-mnf 11296  df-xr 11297  df-ltxr 11298  df-le 11299  df-ioo 13388
This theorem is referenced by:  volioofmpt  45950  voliooicof  45952
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