Users' Mathboxes Mathbox for Glauco Siliprandi < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  fvvolicof Structured version   Visualization version   GIF version

Theorem fvvolicof 39512
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 6005 . . . 4 (𝐹:𝐴⟶(ℝ* × ℝ*) → Fun 𝐹)
31, 2syl 17 . . 3 (𝜑 → Fun 𝐹)
4 fvvolicof.x . . . 4 (𝜑𝑋𝐴)
5 fdm 6008 . . . . . 6 (𝐹:𝐴⟶(ℝ* × ℝ*) → dom 𝐹 = 𝐴)
61, 5syl 17 . . . . 5 (𝜑 → dom 𝐹 = 𝐴)
76eqcomd 2627 . . . 4 (𝜑𝐴 = dom 𝐹)
84, 7eleqtrd 2700 . . 3 (𝜑𝑋 ∈ dom 𝐹)
9 fvco 6231 . . 3 ((Fun 𝐹𝑋 ∈ dom 𝐹) → (((vol ∘ [,)) ∘ 𝐹)‘𝑋) = ((vol ∘ [,))‘(𝐹𝑋)))
103, 8, 9syl2anc 692 . 2 (𝜑 → (((vol ∘ [,)) ∘ 𝐹)‘𝑋) = ((vol ∘ [,))‘(𝐹𝑋)))
11 icof 38882 . . . . 5 [,):(ℝ* × ℝ*)⟶𝒫 ℝ*
12 ffun 6005 . . . . 5 ([,):(ℝ* × ℝ*)⟶𝒫 ℝ* → Fun [,))
1311, 12ax-mp 5 . . . 4 Fun [,)
1413a1i 11 . . 3 (𝜑 → Fun [,))
151, 4ffvelrnd 6316 . . . 4 (𝜑 → (𝐹𝑋) ∈ (ℝ* × ℝ*))
1611fdmi 6009 . . . 4 dom [,) = (ℝ* × ℝ*)
1715, 16syl6eleqr 2709 . . 3 (𝜑 → (𝐹𝑋) ∈ dom [,))
18 fvco 6231 . . 3 ((Fun [,) ∧ (𝐹𝑋) ∈ dom [,)) → ((vol ∘ [,))‘(𝐹𝑋)) = (vol‘([,)‘(𝐹𝑋))))
1914, 17, 18syl2anc 692 . 2 (𝜑 → ((vol ∘ [,))‘(𝐹𝑋)) = (vol‘([,)‘(𝐹𝑋))))
20 df-ov 6607 . . . . 5 ((1st ‘(𝐹𝑋))[,)(2nd ‘(𝐹𝑋))) = ([,)‘⟨(1st ‘(𝐹𝑋)), (2nd ‘(𝐹𝑋))⟩)
2120a1i 11 . . . 4 (𝜑 → ((1st ‘(𝐹𝑋))[,)(2nd ‘(𝐹𝑋))) = ([,)‘⟨(1st ‘(𝐹𝑋)), (2nd ‘(𝐹𝑋))⟩))
22 1st2nd2 7150 . . . . . . 7 ((𝐹𝑋) ∈ (ℝ* × ℝ*) → (𝐹𝑋) = ⟨(1st ‘(𝐹𝑋)), (2nd ‘(𝐹𝑋))⟩)
2315, 22syl 17 . . . . . 6 (𝜑 → (𝐹𝑋) = ⟨(1st ‘(𝐹𝑋)), (2nd ‘(𝐹𝑋))⟩)
2423eqcomd 2627 . . . . 5 (𝜑 → ⟨(1st ‘(𝐹𝑋)), (2nd ‘(𝐹𝑋))⟩ = (𝐹𝑋))
2524fveq2d 6152 . . . 4 (𝜑 → ([,)‘⟨(1st ‘(𝐹𝑋)), (2nd ‘(𝐹𝑋))⟩) = ([,)‘(𝐹𝑋)))
2621, 25eqtr2d 2656 . . 3 (𝜑 → ([,)‘(𝐹𝑋)) = ((1st ‘(𝐹𝑋))[,)(2nd ‘(𝐹𝑋))))
2726fveq2d 6152 . 2 (𝜑 → (vol‘([,)‘(𝐹𝑋))) = (vol‘((1st ‘(𝐹𝑋))[,)(2nd ‘(𝐹𝑋)))))
2810, 19, 273eqtrd 2659 1 (𝜑 → (((vol ∘ [,)) ∘ 𝐹)‘𝑋) = (vol‘((1st ‘(𝐹𝑋))[,)(2nd ‘(𝐹𝑋)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  wcel 1987  𝒫 cpw 4130  cop 4154   × cxp 5072  dom cdm 5074  ccom 5078  Fun wfun 5841  wf 5843  cfv 5847  (class class class)co 6604  1st c1st 7111  2nd c2nd 7112  *cxr 10017  [,)cico 12119  volcvol 23139
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-1st 7113  df-2nd 7114  df-xr 10022  df-ico 12123
This theorem is referenced by:  voliooicof  39517  volicofmpt  39518
  Copyright terms: Public domain W3C validator