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Theorem fvvolioof 40937
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 6258 . . 3 (𝜑 → Fun 𝐹)
3 fvvolioof.x . . . 4 (𝜑𝑋𝐴)
41fdmd 6263 . . . . 5 (𝜑 → dom 𝐹 = 𝐴)
54eqcomd 2803 . . . 4 (𝜑𝐴 = dom 𝐹)
63, 5eleqtrd 2878 . . 3 (𝜑𝑋 ∈ dom 𝐹)
7 fvco 6497 . . 3 ((Fun 𝐹𝑋 ∈ dom 𝐹) → (((vol ∘ (,)) ∘ 𝐹)‘𝑋) = ((vol ∘ (,))‘(𝐹𝑋)))
82, 6, 7syl2anc 580 . 2 (𝜑 → (((vol ∘ (,)) ∘ 𝐹)‘𝑋) = ((vol ∘ (,))‘(𝐹𝑋)))
9 ioof 12517 . . . . 5 (,):(ℝ* × ℝ*)⟶𝒫 ℝ
10 ffun 6257 . . . . 5 ((,):(ℝ* × ℝ*)⟶𝒫 ℝ → Fun (,))
119, 10ax-mp 5 . . . 4 Fun (,)
1211a1i 11 . . 3 (𝜑 → Fun (,))
131, 3ffvelrnd 6584 . . . 4 (𝜑 → (𝐹𝑋) ∈ (ℝ* × ℝ*))
149fdmi 6264 . . . 4 dom (,) = (ℝ* × ℝ*)
1513, 14syl6eleqr 2887 . . 3 (𝜑 → (𝐹𝑋) ∈ dom (,))
16 fvco 6497 . . 3 ((Fun (,) ∧ (𝐹𝑋) ∈ dom (,)) → ((vol ∘ (,))‘(𝐹𝑋)) = (vol‘((,)‘(𝐹𝑋))))
1712, 15, 16syl2anc 580 . 2 (𝜑 → ((vol ∘ (,))‘(𝐹𝑋)) = (vol‘((,)‘(𝐹𝑋))))
18 df-ov 6879 . . . . 5 ((1st ‘(𝐹𝑋))(,)(2nd ‘(𝐹𝑋))) = ((,)‘⟨(1st ‘(𝐹𝑋)), (2nd ‘(𝐹𝑋))⟩)
1918a1i 11 . . . 4 (𝜑 → ((1st ‘(𝐹𝑋))(,)(2nd ‘(𝐹𝑋))) = ((,)‘⟨(1st ‘(𝐹𝑋)), (2nd ‘(𝐹𝑋))⟩))
20 1st2nd2 7438 . . . . . . 7 ((𝐹𝑋) ∈ (ℝ* × ℝ*) → (𝐹𝑋) = ⟨(1st ‘(𝐹𝑋)), (2nd ‘(𝐹𝑋))⟩)
2113, 20syl 17 . . . . . 6 (𝜑 → (𝐹𝑋) = ⟨(1st ‘(𝐹𝑋)), (2nd ‘(𝐹𝑋))⟩)
2221eqcomd 2803 . . . . 5 (𝜑 → ⟨(1st ‘(𝐹𝑋)), (2nd ‘(𝐹𝑋))⟩ = (𝐹𝑋))
2322fveq2d 6413 . . . 4 (𝜑 → ((,)‘⟨(1st ‘(𝐹𝑋)), (2nd ‘(𝐹𝑋))⟩) = ((,)‘(𝐹𝑋)))
2419, 23eqtr2d 2832 . . 3 (𝜑 → ((,)‘(𝐹𝑋)) = ((1st ‘(𝐹𝑋))(,)(2nd ‘(𝐹𝑋))))
2524fveq2d 6413 . 2 (𝜑 → (vol‘((,)‘(𝐹𝑋))) = (vol‘((1st ‘(𝐹𝑋))(,)(2nd ‘(𝐹𝑋)))))
268, 17, 253eqtrd 2835 1 (𝜑 → (((vol ∘ (,)) ∘ 𝐹)‘𝑋) = (vol‘((1st ‘(𝐹𝑋))(,)(2nd ‘(𝐹𝑋)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1653  wcel 2157  𝒫 cpw 4347  cop 4372   × cxp 5308  dom cdm 5310  ccom 5314  Fun wfun 6093  wf 6095  cfv 6099  (class class class)co 6876  1st c1st 7397  2nd c2nd 7398  cr 10221  *cxr 10360  (,)cioo 12420  volcvol 23568
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2354  ax-ext 2775  ax-sep 4973  ax-nul 4981  ax-pow 5033  ax-pr 5095  ax-un 7181  ax-cnex 10278  ax-resscn 10279  ax-pre-lttri 10296  ax-pre-lttrn 10297
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3or 1109  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2590  df-eu 2607  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-ne 2970  df-nel 3073  df-ral 3092  df-rex 3093  df-rab 3096  df-v 3385  df-sbc 3632  df-csb 3727  df-dif 3770  df-un 3772  df-in 3774  df-ss 3781  df-nul 4114  df-if 4276  df-pw 4349  df-sn 4367  df-pr 4369  df-op 4373  df-uni 4627  df-iun 4710  df-br 4842  df-opab 4904  df-mpt 4921  df-id 5218  df-po 5231  df-so 5232  df-xp 5316  df-rel 5317  df-cnv 5318  df-co 5319  df-dm 5320  df-rn 5321  df-res 5322  df-ima 5323  df-iota 6062  df-fun 6101  df-fn 6102  df-f 6103  df-f1 6104  df-fo 6105  df-f1o 6106  df-fv 6107  df-ov 6879  df-oprab 6880  df-mpt2 6881  df-1st 7399  df-2nd 7400  df-er 7980  df-en 8194  df-dom 8195  df-sdom 8196  df-pnf 10363  df-mnf 10364  df-xr 10365  df-ltxr 10366  df-le 10367  df-ioo 12424
This theorem is referenced by:  volioofmpt  40942  voliooicof  40944
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