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Theorem fvvolicof 45612
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*))
21ffund 6732 . . 3 (𝜑 → Fun 𝐹)
3 fvvolicof.x . . . 4 (𝜑𝑋𝐴)
41fdmd 6738 . . . . 5 (𝜑 → dom 𝐹 = 𝐴)
54eqcomd 2732 . . . 4 (𝜑𝐴 = dom 𝐹)
63, 5eleqtrd 2828 . . 3 (𝜑𝑋 ∈ dom 𝐹)
7 fvco 7000 . . 3 ((Fun 𝐹𝑋 ∈ dom 𝐹) → (((vol ∘ [,)) ∘ 𝐹)‘𝑋) = ((vol ∘ [,))‘(𝐹𝑋)))
82, 6, 7syl2anc 582 . 2 (𝜑 → (((vol ∘ [,)) ∘ 𝐹)‘𝑋) = ((vol ∘ [,))‘(𝐹𝑋)))
9 icof 44826 . . . . 5 [,):(ℝ* × ℝ*)⟶𝒫 ℝ*
10 ffun 6731 . . . . 5 ([,):(ℝ* × ℝ*)⟶𝒫 ℝ* → Fun [,))
119, 10ax-mp 5 . . . 4 Fun [,)
1211a1i 11 . . 3 (𝜑 → Fun [,))
131, 3ffvelcdmd 7099 . . . 4 (𝜑 → (𝐹𝑋) ∈ (ℝ* × ℝ*))
149fdmi 6739 . . . 4 dom [,) = (ℝ* × ℝ*)
1513, 14eleqtrrdi 2837 . . 3 (𝜑 → (𝐹𝑋) ∈ dom [,))
16 fvco 7000 . . 3 ((Fun [,) ∧ (𝐹𝑋) ∈ dom [,)) → ((vol ∘ [,))‘(𝐹𝑋)) = (vol‘([,)‘(𝐹𝑋))))
1712, 15, 16syl2anc 582 . 2 (𝜑 → ((vol ∘ [,))‘(𝐹𝑋)) = (vol‘([,)‘(𝐹𝑋))))
18 df-ov 7427 . . . . 5 ((1st ‘(𝐹𝑋))[,)(2nd ‘(𝐹𝑋))) = ([,)‘⟨(1st ‘(𝐹𝑋)), (2nd ‘(𝐹𝑋))⟩)
1918a1i 11 . . . 4 (𝜑 → ((1st ‘(𝐹𝑋))[,)(2nd ‘(𝐹𝑋))) = ([,)‘⟨(1st ‘(𝐹𝑋)), (2nd ‘(𝐹𝑋))⟩))
20 1st2nd2 8042 . . . . . . 7 ((𝐹𝑋) ∈ (ℝ* × ℝ*) → (𝐹𝑋) = ⟨(1st ‘(𝐹𝑋)), (2nd ‘(𝐹𝑋))⟩)
2113, 20syl 17 . . . . . 6 (𝜑 → (𝐹𝑋) = ⟨(1st ‘(𝐹𝑋)), (2nd ‘(𝐹𝑋))⟩)
2221eqcomd 2732 . . . . 5 (𝜑 → ⟨(1st ‘(𝐹𝑋)), (2nd ‘(𝐹𝑋))⟩ = (𝐹𝑋))
2322fveq2d 6905 . . . 4 (𝜑 → ([,)‘⟨(1st ‘(𝐹𝑋)), (2nd ‘(𝐹𝑋))⟩) = ([,)‘(𝐹𝑋)))
2419, 23eqtr2d 2767 . . 3 (𝜑 → ([,)‘(𝐹𝑋)) = ((1st ‘(𝐹𝑋))[,)(2nd ‘(𝐹𝑋))))
2524fveq2d 6905 . 2 (𝜑 → (vol‘([,)‘(𝐹𝑋))) = (vol‘((1st ‘(𝐹𝑋))[,)(2nd ‘(𝐹𝑋)))))
268, 17, 253eqtrd 2770 1 (𝜑 → (((vol ∘ [,)) ∘ 𝐹)‘𝑋) = (vol‘((1st ‘(𝐹𝑋))[,)(2nd ‘(𝐹𝑋)))))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1534  wcel 2099  𝒫 cpw 4607  cop 4639   × cxp 5680  dom cdm 5682  ccom 5686  Fun wfun 6548  wf 6550  cfv 6554  (class class class)co 7424  1st c1st 8001  2nd c2nd 8002  *cxr 11297  [,)cico 13380  volcvol 25483
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pr 5433  ax-un 7746  ax-cnex 11214  ax-resscn 11215
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-fv 6562  df-ov 7427  df-oprab 7428  df-mpo 7429  df-1st 8003  df-2nd 8004  df-xr 11302  df-ico 13384
This theorem is referenced by:  voliooicof  45617  volicofmpt  45618
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