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Theorem fvvolicof 45379
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 (πœ‘ β†’ 𝐹:𝐴⟢(ℝ* Γ— ℝ*))
fvvolicof.x (πœ‘ β†’ 𝑋 ∈ 𝐴)
Assertion
Ref Expression
fvvolicof (πœ‘ β†’ (((vol ∘ [,)) ∘ 𝐹)β€˜π‘‹) = (volβ€˜((1st β€˜(πΉβ€˜π‘‹))[,)(2nd β€˜(πΉβ€˜π‘‹)))))

Proof of Theorem fvvolicof
StepHypRef Expression
1 fvvolicof.f . . . 4 (πœ‘ β†’ 𝐹:𝐴⟢(ℝ* Γ— ℝ*))
21ffund 6726 . . 3 (πœ‘ β†’ Fun 𝐹)
3 fvvolicof.x . . . 4 (πœ‘ β†’ 𝑋 ∈ 𝐴)
41fdmd 6733 . . . . 5 (πœ‘ β†’ dom 𝐹 = 𝐴)
54eqcomd 2734 . . . 4 (πœ‘ β†’ 𝐴 = dom 𝐹)
63, 5eleqtrd 2831 . . 3 (πœ‘ β†’ 𝑋 ∈ dom 𝐹)
7 fvco 6996 . . 3 ((Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹) β†’ (((vol ∘ [,)) ∘ 𝐹)β€˜π‘‹) = ((vol ∘ [,))β€˜(πΉβ€˜π‘‹)))
82, 6, 7syl2anc 583 . 2 (πœ‘ β†’ (((vol ∘ [,)) ∘ 𝐹)β€˜π‘‹) = ((vol ∘ [,))β€˜(πΉβ€˜π‘‹)))
9 icof 44592 . . . . 5 [,):(ℝ* Γ— ℝ*)βŸΆπ’« ℝ*
10 ffun 6725 . . . . 5 ([,):(ℝ* Γ— ℝ*)βŸΆπ’« ℝ* β†’ Fun [,))
119, 10ax-mp 5 . . . 4 Fun [,)
1211a1i 11 . . 3 (πœ‘ β†’ Fun [,))
131, 3ffvelcdmd 7095 . . . 4 (πœ‘ β†’ (πΉβ€˜π‘‹) ∈ (ℝ* Γ— ℝ*))
149fdmi 6734 . . . 4 dom [,) = (ℝ* Γ— ℝ*)
1513, 14eleqtrrdi 2840 . . 3 (πœ‘ β†’ (πΉβ€˜π‘‹) ∈ dom [,))
16 fvco 6996 . . 3 ((Fun [,) ∧ (πΉβ€˜π‘‹) ∈ dom [,)) β†’ ((vol ∘ [,))β€˜(πΉβ€˜π‘‹)) = (volβ€˜([,)β€˜(πΉβ€˜π‘‹))))
1712, 15, 16syl2anc 583 . 2 (πœ‘ β†’ ((vol ∘ [,))β€˜(πΉβ€˜π‘‹)) = (volβ€˜([,)β€˜(πΉβ€˜π‘‹))))
18 df-ov 7423 . . . . 5 ((1st β€˜(πΉβ€˜π‘‹))[,)(2nd β€˜(πΉβ€˜π‘‹))) = ([,)β€˜βŸ¨(1st β€˜(πΉβ€˜π‘‹)), (2nd β€˜(πΉβ€˜π‘‹))⟩)
1918a1i 11 . . . 4 (πœ‘ β†’ ((1st β€˜(πΉβ€˜π‘‹))[,)(2nd β€˜(πΉβ€˜π‘‹))) = ([,)β€˜βŸ¨(1st β€˜(πΉβ€˜π‘‹)), (2nd β€˜(πΉβ€˜π‘‹))⟩))
20 1st2nd2 8032 . . . . . . 7 ((πΉβ€˜π‘‹) ∈ (ℝ* Γ— ℝ*) β†’ (πΉβ€˜π‘‹) = ⟨(1st β€˜(πΉβ€˜π‘‹)), (2nd β€˜(πΉβ€˜π‘‹))⟩)
2113, 20syl 17 . . . . . 6 (πœ‘ β†’ (πΉβ€˜π‘‹) = ⟨(1st β€˜(πΉβ€˜π‘‹)), (2nd β€˜(πΉβ€˜π‘‹))⟩)
2221eqcomd 2734 . . . . 5 (πœ‘ β†’ ⟨(1st β€˜(πΉβ€˜π‘‹)), (2nd β€˜(πΉβ€˜π‘‹))⟩ = (πΉβ€˜π‘‹))
2322fveq2d 6901 . . . 4 (πœ‘ β†’ ([,)β€˜βŸ¨(1st β€˜(πΉβ€˜π‘‹)), (2nd β€˜(πΉβ€˜π‘‹))⟩) = ([,)β€˜(πΉβ€˜π‘‹)))
2419, 23eqtr2d 2769 . . 3 (πœ‘ β†’ ([,)β€˜(πΉβ€˜π‘‹)) = ((1st β€˜(πΉβ€˜π‘‹))[,)(2nd β€˜(πΉβ€˜π‘‹))))
2524fveq2d 6901 . 2 (πœ‘ β†’ (volβ€˜([,)β€˜(πΉβ€˜π‘‹))) = (volβ€˜((1st β€˜(πΉβ€˜π‘‹))[,)(2nd β€˜(πΉβ€˜π‘‹)))))
268, 17, 253eqtrd 2772 1 (πœ‘ β†’ (((vol ∘ [,)) ∘ 𝐹)β€˜π‘‹) = (volβ€˜((1st β€˜(πΉβ€˜π‘‹))[,)(2nd β€˜(πΉβ€˜π‘‹)))))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   = wceq 1534   ∈ wcel 2099  π’« cpw 4603  βŸ¨cop 4635   Γ— cxp 5676  dom cdm 5678   ∘ ccom 5682  Fun wfun 6542  βŸΆwf 6544  β€˜cfv 6548  (class class class)co 7420  1st c1st 7991  2nd c2nd 7992  β„*cxr 11278  [,)cico 13359  volcvol 25405
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 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429  ax-un 7740  ax-cnex 11195  ax-resscn 11196
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-fv 6556  df-ov 7423  df-oprab 7424  df-mpo 7425  df-1st 7993  df-2nd 7994  df-xr 11283  df-ico 13363
This theorem is referenced by:  voliooicof  45384  volicofmpt  45385
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