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Theorem fvvolioof 42422
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 6494 . . 3 (𝜑 → Fun 𝐹)
3 fvvolioof.x . . . 4 (𝜑𝑋𝐴)
41fdmd 6499 . . . . 5 (𝜑 → dom 𝐹 = 𝐴)
54eqcomd 2826 . . . 4 (𝜑𝐴 = dom 𝐹)
63, 5eleqtrd 2913 . . 3 (𝜑𝑋 ∈ dom 𝐹)
7 fvco 6735 . . 3 ((Fun 𝐹𝑋 ∈ dom 𝐹) → (((vol ∘ (,)) ∘ 𝐹)‘𝑋) = ((vol ∘ (,))‘(𝐹𝑋)))
82, 6, 7syl2anc 586 . 2 (𝜑 → (((vol ∘ (,)) ∘ 𝐹)‘𝑋) = ((vol ∘ (,))‘(𝐹𝑋)))
9 ioof 12816 . . . . 5 (,):(ℝ* × ℝ*)⟶𝒫 ℝ
10 ffun 6493 . . . . 5 ((,):(ℝ* × ℝ*)⟶𝒫 ℝ → Fun (,))
119, 10ax-mp 5 . . . 4 Fun (,)
1211a1i 11 . . 3 (𝜑 → Fun (,))
131, 3ffvelrnd 6828 . . . 4 (𝜑 → (𝐹𝑋) ∈ (ℝ* × ℝ*))
149fdmi 6500 . . . 4 dom (,) = (ℝ* × ℝ*)
1513, 14eleqtrrdi 2922 . . 3 (𝜑 → (𝐹𝑋) ∈ dom (,))
16 fvco 6735 . . 3 ((Fun (,) ∧ (𝐹𝑋) ∈ dom (,)) → ((vol ∘ (,))‘(𝐹𝑋)) = (vol‘((,)‘(𝐹𝑋))))
1712, 15, 16syl2anc 586 . 2 (𝜑 → ((vol ∘ (,))‘(𝐹𝑋)) = (vol‘((,)‘(𝐹𝑋))))
18 df-ov 7136 . . . . 5 ((1st ‘(𝐹𝑋))(,)(2nd ‘(𝐹𝑋))) = ((,)‘⟨(1st ‘(𝐹𝑋)), (2nd ‘(𝐹𝑋))⟩)
1918a1i 11 . . . 4 (𝜑 → ((1st ‘(𝐹𝑋))(,)(2nd ‘(𝐹𝑋))) = ((,)‘⟨(1st ‘(𝐹𝑋)), (2nd ‘(𝐹𝑋))⟩))
20 1st2nd2 7706 . . . . . . 7 ((𝐹𝑋) ∈ (ℝ* × ℝ*) → (𝐹𝑋) = ⟨(1st ‘(𝐹𝑋)), (2nd ‘(𝐹𝑋))⟩)
2113, 20syl 17 . . . . . 6 (𝜑 → (𝐹𝑋) = ⟨(1st ‘(𝐹𝑋)), (2nd ‘(𝐹𝑋))⟩)
2221eqcomd 2826 . . . . 5 (𝜑 → ⟨(1st ‘(𝐹𝑋)), (2nd ‘(𝐹𝑋))⟩ = (𝐹𝑋))
2322fveq2d 6650 . . . 4 (𝜑 → ((,)‘⟨(1st ‘(𝐹𝑋)), (2nd ‘(𝐹𝑋))⟩) = ((,)‘(𝐹𝑋)))
2419, 23eqtr2d 2856 . . 3 (𝜑 → ((,)‘(𝐹𝑋)) = ((1st ‘(𝐹𝑋))(,)(2nd ‘(𝐹𝑋))))
2524fveq2d 6650 . 2 (𝜑 → (vol‘((,)‘(𝐹𝑋))) = (vol‘((1st ‘(𝐹𝑋))(,)(2nd ‘(𝐹𝑋)))))
268, 17, 253eqtrd 2859 1 (𝜑 → (((vol ∘ (,)) ∘ 𝐹)‘𝑋) = (vol‘((1st ‘(𝐹𝑋))(,)(2nd ‘(𝐹𝑋)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2114  𝒫 cpw 4515  cop 4549   × cxp 5529  dom cdm 5531  ccom 5535  Fun wfun 6325  wf 6327  cfv 6331  (class class class)co 7133  1st c1st 7665  2nd c2nd 7666  cr 10514  *cxr 10652  (,)cioo 12717  volcvol 24046
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-sep 5179  ax-nul 5186  ax-pow 5242  ax-pr 5306  ax-un 7439  ax-cnex 10571  ax-resscn 10572  ax-pre-lttri 10589  ax-pre-lttrn 10590
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-nel 3111  df-ral 3130  df-rex 3131  df-rab 3134  df-v 3475  df-sbc 3753  df-csb 3861  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4270  df-if 4444  df-pw 4517  df-sn 4544  df-pr 4546  df-op 4550  df-uni 4815  df-iun 4897  df-br 5043  df-opab 5105  df-mpt 5123  df-id 5436  df-po 5450  df-so 5451  df-xp 5537  df-rel 5538  df-cnv 5539  df-co 5540  df-dm 5541  df-rn 5542  df-res 5543  df-ima 5544  df-iota 6290  df-fun 6333  df-fn 6334  df-f 6335  df-f1 6336  df-fo 6337  df-f1o 6338  df-fv 6339  df-ov 7136  df-oprab 7137  df-mpo 7138  df-1st 7667  df-2nd 7668  df-er 8267  df-en 8488  df-dom 8489  df-sdom 8490  df-pnf 10655  df-mnf 10656  df-xr 10657  df-ltxr 10658  df-le 10659  df-ioo 12721
This theorem is referenced by:  volioofmpt  42427  voliooicof  42429
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