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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > fvvolicof | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
fvvolicof.f | ⊢ (𝜑 → 𝐹:𝐴⟶(ℝ* × ℝ*)) |
fvvolicof.x | ⊢ (𝜑 → 𝑋 ∈ 𝐴) |
Ref | Expression |
---|---|
fvvolicof | ⊢ (𝜑 → (((vol ∘ [,)) ∘ 𝐹)‘𝑋) = (vol‘((1st ‘(𝐹‘𝑋))[,)(2nd ‘(𝐹‘𝑋))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvvolicof.f | . . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶(ℝ* × ℝ*)) | |
2 | 1 | ffund 6732 | . . 3 ⊢ (𝜑 → Fun 𝐹) |
3 | fvvolicof.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐴) | |
4 | 1 | fdmd 6738 | . . . . 5 ⊢ (𝜑 → dom 𝐹 = 𝐴) |
5 | 4 | eqcomd 2732 | . . . 4 ⊢ (𝜑 → 𝐴 = dom 𝐹) |
6 | 3, 5 | eleqtrd 2828 | . . 3 ⊢ (𝜑 → 𝑋 ∈ dom 𝐹) |
7 | fvco 7000 | . . 3 ⊢ ((Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹) → (((vol ∘ [,)) ∘ 𝐹)‘𝑋) = ((vol ∘ [,))‘(𝐹‘𝑋))) | |
8 | 2, 6, 7 | syl2anc 582 | . 2 ⊢ (𝜑 → (((vol ∘ [,)) ∘ 𝐹)‘𝑋) = ((vol ∘ [,))‘(𝐹‘𝑋))) |
9 | icof 44826 | . . . . 5 ⊢ [,):(ℝ* × ℝ*)⟶𝒫 ℝ* | |
10 | ffun 6731 | . . . . 5 ⊢ ([,):(ℝ* × ℝ*)⟶𝒫 ℝ* → Fun [,)) | |
11 | 9, 10 | ax-mp 5 | . . . 4 ⊢ Fun [,) |
12 | 11 | a1i 11 | . . 3 ⊢ (𝜑 → Fun [,)) |
13 | 1, 3 | ffvelcdmd 7099 | . . . 4 ⊢ (𝜑 → (𝐹‘𝑋) ∈ (ℝ* × ℝ*)) |
14 | 9 | fdmi 6739 | . . . 4 ⊢ dom [,) = (ℝ* × ℝ*) |
15 | 13, 14 | eleqtrrdi 2837 | . . 3 ⊢ (𝜑 → (𝐹‘𝑋) ∈ dom [,)) |
16 | fvco 7000 | . . 3 ⊢ ((Fun [,) ∧ (𝐹‘𝑋) ∈ dom [,)) → ((vol ∘ [,))‘(𝐹‘𝑋)) = (vol‘([,)‘(𝐹‘𝑋)))) | |
17 | 12, 15, 16 | syl2anc 582 | . 2 ⊢ (𝜑 → ((vol ∘ [,))‘(𝐹‘𝑋)) = (vol‘([,)‘(𝐹‘𝑋)))) |
18 | df-ov 7427 | . . . . 5 ⊢ ((1st ‘(𝐹‘𝑋))[,)(2nd ‘(𝐹‘𝑋))) = ([,)‘〈(1st ‘(𝐹‘𝑋)), (2nd ‘(𝐹‘𝑋))〉) | |
19 | 18 | a1i 11 | . . . 4 ⊢ (𝜑 → ((1st ‘(𝐹‘𝑋))[,)(2nd ‘(𝐹‘𝑋))) = ([,)‘〈(1st ‘(𝐹‘𝑋)), (2nd ‘(𝐹‘𝑋))〉)) |
20 | 1st2nd2 8042 | . . . . . . 7 ⊢ ((𝐹‘𝑋) ∈ (ℝ* × ℝ*) → (𝐹‘𝑋) = 〈(1st ‘(𝐹‘𝑋)), (2nd ‘(𝐹‘𝑋))〉) | |
21 | 13, 20 | syl 17 | . . . . . 6 ⊢ (𝜑 → (𝐹‘𝑋) = 〈(1st ‘(𝐹‘𝑋)), (2nd ‘(𝐹‘𝑋))〉) |
22 | 21 | eqcomd 2732 | . . . . 5 ⊢ (𝜑 → 〈(1st ‘(𝐹‘𝑋)), (2nd ‘(𝐹‘𝑋))〉 = (𝐹‘𝑋)) |
23 | 22 | fveq2d 6905 | . . . 4 ⊢ (𝜑 → ([,)‘〈(1st ‘(𝐹‘𝑋)), (2nd ‘(𝐹‘𝑋))〉) = ([,)‘(𝐹‘𝑋))) |
24 | 19, 23 | eqtr2d 2767 | . . 3 ⊢ (𝜑 → ([,)‘(𝐹‘𝑋)) = ((1st ‘(𝐹‘𝑋))[,)(2nd ‘(𝐹‘𝑋)))) |
25 | 24 | fveq2d 6905 | . 2 ⊢ (𝜑 → (vol‘([,)‘(𝐹‘𝑋))) = (vol‘((1st ‘(𝐹‘𝑋))[,)(2nd ‘(𝐹‘𝑋))))) |
26 | 8, 17, 25 | 3eqtrd 2770 | 1 ⊢ (𝜑 → (((vol ∘ [,)) ∘ 𝐹)‘𝑋) = (vol‘((1st ‘(𝐹‘𝑋))[,)(2nd ‘(𝐹‘𝑋))))) |
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 ℝ*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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