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Theorem fvvolicof 40719
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 (𝜑𝐹:𝐴⟶(ℝ* × ℝ*))
fvvolicof.x (𝜑𝑋𝐴)
Assertion
Ref Expression
fvvolicof (𝜑 → (((vol ∘ [,)) ∘ 𝐹)‘𝑋) = (vol‘((1st ‘(𝐹𝑋))[,)(2nd ‘(𝐹𝑋)))))

Proof of Theorem fvvolicof
StepHypRef Expression
1 fvvolicof.f . . . 4 (𝜑𝐹:𝐴⟶(ℝ* × ℝ*))
2 ffun 6188 . . . 4 (𝐹:𝐴⟶(ℝ* × ℝ*) → Fun 𝐹)
31, 2syl 17 . . 3 (𝜑 → Fun 𝐹)
4 fvvolicof.x . . . 4 (𝜑𝑋𝐴)
5 fdm 6191 . . . . . 6 (𝐹:𝐴⟶(ℝ* × ℝ*) → dom 𝐹 = 𝐴)
61, 5syl 17 . . . . 5 (𝜑 → dom 𝐹 = 𝐴)
76eqcomd 2776 . . . 4 (𝜑𝐴 = dom 𝐹)
84, 7eleqtrd 2851 . . 3 (𝜑𝑋 ∈ dom 𝐹)
9 fvco 6416 . . 3 ((Fun 𝐹𝑋 ∈ dom 𝐹) → (((vol ∘ [,)) ∘ 𝐹)‘𝑋) = ((vol ∘ [,))‘(𝐹𝑋)))
103, 8, 9syl2anc 565 . 2 (𝜑 → (((vol ∘ [,)) ∘ 𝐹)‘𝑋) = ((vol ∘ [,))‘(𝐹𝑋)))
11 icof 39923 . . . . 5 [,):(ℝ* × ℝ*)⟶𝒫 ℝ*
12 ffun 6188 . . . . 5 ([,):(ℝ* × ℝ*)⟶𝒫 ℝ* → Fun [,))
1311, 12ax-mp 5 . . . 4 Fun [,)
1413a1i 11 . . 3 (𝜑 → Fun [,))
151, 4ffvelrnd 6503 . . . 4 (𝜑 → (𝐹𝑋) ∈ (ℝ* × ℝ*))
1611fdmi 6192 . . . 4 dom [,) = (ℝ* × ℝ*)
1715, 16syl6eleqr 2860 . . 3 (𝜑 → (𝐹𝑋) ∈ dom [,))
18 fvco 6416 . . 3 ((Fun [,) ∧ (𝐹𝑋) ∈ dom [,)) → ((vol ∘ [,))‘(𝐹𝑋)) = (vol‘([,)‘(𝐹𝑋))))
1914, 17, 18syl2anc 565 . 2 (𝜑 → ((vol ∘ [,))‘(𝐹𝑋)) = (vol‘([,)‘(𝐹𝑋))))
20 df-ov 6795 . . . . 5 ((1st ‘(𝐹𝑋))[,)(2nd ‘(𝐹𝑋))) = ([,)‘⟨(1st ‘(𝐹𝑋)), (2nd ‘(𝐹𝑋))⟩)
2120a1i 11 . . . 4 (𝜑 → ((1st ‘(𝐹𝑋))[,)(2nd ‘(𝐹𝑋))) = ([,)‘⟨(1st ‘(𝐹𝑋)), (2nd ‘(𝐹𝑋))⟩))
22 1st2nd2 7353 . . . . . . 7 ((𝐹𝑋) ∈ (ℝ* × ℝ*) → (𝐹𝑋) = ⟨(1st ‘(𝐹𝑋)), (2nd ‘(𝐹𝑋))⟩)
2315, 22syl 17 . . . . . 6 (𝜑 → (𝐹𝑋) = ⟨(1st ‘(𝐹𝑋)), (2nd ‘(𝐹𝑋))⟩)
2423eqcomd 2776 . . . . 5 (𝜑 → ⟨(1st ‘(𝐹𝑋)), (2nd ‘(𝐹𝑋))⟩ = (𝐹𝑋))
2524fveq2d 6336 . . . 4 (𝜑 → ([,)‘⟨(1st ‘(𝐹𝑋)), (2nd ‘(𝐹𝑋))⟩) = ([,)‘(𝐹𝑋)))
2621, 25eqtr2d 2805 . . 3 (𝜑 → ([,)‘(𝐹𝑋)) = ((1st ‘(𝐹𝑋))[,)(2nd ‘(𝐹𝑋))))
2726fveq2d 6336 . 2 (𝜑 → (vol‘([,)‘(𝐹𝑋))) = (vol‘((1st ‘(𝐹𝑋))[,)(2nd ‘(𝐹𝑋)))))
2810, 19, 273eqtrd 2808 1 (𝜑 → (((vol ∘ [,)) ∘ 𝐹)‘𝑋) = (vol‘((1st ‘(𝐹𝑋))[,)(2nd ‘(𝐹𝑋)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1630  wcel 2144  𝒫 cpw 4295  cop 4320   × cxp 5247  dom cdm 5249  ccom 5253  Fun wfun 6025  wf 6027  cfv 6031  (class class class)co 6792  1st c1st 7312  2nd c2nd 7313  *cxr 10274  [,)cico 12381  volcvol 23450
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095  ax-cnex 10193  ax-resscn 10194
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-fv 6039  df-ov 6795  df-oprab 6796  df-mpt2 6797  df-1st 7314  df-2nd 7315  df-xr 10279  df-ico 12385
This theorem is referenced by:  voliooicof  40724  volicofmpt  40725
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