Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > fvvolioof | Structured version Visualization version GIF version |
Description: The function value of the Lebesgue measure of an open interval composed with a function. (Contributed by Glauco Siliprandi, 3-Mar-2021.) |
Ref | Expression |
---|---|
fvvolioof.f | ⊢ (𝜑 → 𝐹:𝐴⟶(ℝ* × ℝ*)) |
fvvolioof.x | ⊢ (𝜑 → 𝑋 ∈ 𝐴) |
Ref | Expression |
---|---|
fvvolioof | ⊢ (𝜑 → (((vol ∘ (,)) ∘ 𝐹)‘𝑋) = (vol‘((1st ‘(𝐹‘𝑋))(,)(2nd ‘(𝐹‘𝑋))))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvvolioof.f | . . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶(ℝ* × ℝ*)) | |
2 | 1 | ffund 6511 | . . 3 ⊢ (𝜑 → Fun 𝐹) |
3 | fvvolioof.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐴) | |
4 | 1 | fdmd 6516 | . . . . 5 ⊢ (𝜑 → dom 𝐹 = 𝐴) |
5 | 4 | eqcomd 2824 | . . . 4 ⊢ (𝜑 → 𝐴 = dom 𝐹) |
6 | 3, 5 | eleqtrd 2912 | . . 3 ⊢ (𝜑 → 𝑋 ∈ dom 𝐹) |
7 | fvco 6752 | . . 3 ⊢ ((Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹) → (((vol ∘ (,)) ∘ 𝐹)‘𝑋) = ((vol ∘ (,))‘(𝐹‘𝑋))) | |
8 | 2, 6, 7 | syl2anc 584 | . 2 ⊢ (𝜑 → (((vol ∘ (,)) ∘ 𝐹)‘𝑋) = ((vol ∘ (,))‘(𝐹‘𝑋))) |
9 | ioof 12823 | . . . . 5 ⊢ (,):(ℝ* × ℝ*)⟶𝒫 ℝ | |
10 | ffun 6510 | . . . . 5 ⊢ ((,):(ℝ* × ℝ*)⟶𝒫 ℝ → Fun (,)) | |
11 | 9, 10 | ax-mp 5 | . . . 4 ⊢ Fun (,) |
12 | 11 | a1i 11 | . . 3 ⊢ (𝜑 → Fun (,)) |
13 | 1, 3 | ffvelrnd 6844 | . . . 4 ⊢ (𝜑 → (𝐹‘𝑋) ∈ (ℝ* × ℝ*)) |
14 | 9 | fdmi 6517 | . . . 4 ⊢ dom (,) = (ℝ* × ℝ*) |
15 | 13, 14 | eleqtrrdi 2921 | . . 3 ⊢ (𝜑 → (𝐹‘𝑋) ∈ dom (,)) |
16 | fvco 6752 | . . 3 ⊢ ((Fun (,) ∧ (𝐹‘𝑋) ∈ dom (,)) → ((vol ∘ (,))‘(𝐹‘𝑋)) = (vol‘((,)‘(𝐹‘𝑋)))) | |
17 | 12, 15, 16 | syl2anc 584 | . 2 ⊢ (𝜑 → ((vol ∘ (,))‘(𝐹‘𝑋)) = (vol‘((,)‘(𝐹‘𝑋)))) |
18 | df-ov 7148 | . . . . 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 7717 | . . . . . . 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 2824 | . . . . 5 ⊢ (𝜑 → 〈(1st ‘(𝐹‘𝑋)), (2nd ‘(𝐹‘𝑋))〉 = (𝐹‘𝑋)) |
23 | 22 | fveq2d 6667 | . . . 4 ⊢ (𝜑 → ((,)‘〈(1st ‘(𝐹‘𝑋)), (2nd ‘(𝐹‘𝑋))〉) = ((,)‘(𝐹‘𝑋))) |
24 | 19, 23 | eqtr2d 2854 | . . 3 ⊢ (𝜑 → ((,)‘(𝐹‘𝑋)) = ((1st ‘(𝐹‘𝑋))(,)(2nd ‘(𝐹‘𝑋)))) |
25 | 24 | fveq2d 6667 | . 2 ⊢ (𝜑 → (vol‘((,)‘(𝐹‘𝑋))) = (vol‘((1st ‘(𝐹‘𝑋))(,)(2nd ‘(𝐹‘𝑋))))) |
26 | 8, 17, 25 | 3eqtrd 2857 | 1 ⊢ (𝜑 → (((vol ∘ (,)) ∘ 𝐹)‘𝑋) = (vol‘((1st ‘(𝐹‘𝑋))(,)(2nd ‘(𝐹‘𝑋))))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 𝒫 cpw 4535 〈cop 4563 × cxp 5546 dom cdm 5548 ∘ ccom 5552 Fun wfun 6342 ⟶wf 6344 ‘cfv 6348 (class class class)co 7145 1st c1st 7676 2nd c2nd 7677 ℝcr 10524 ℝ*cxr 10662 (,)cioo 12726 volcvol 23991 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 ax-cnex 10581 ax-resscn 10582 ax-pre-lttri 10599 ax-pre-lttrn 10600 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-nel 3121 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-po 5467 df-so 5468 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-1st 7678 df-2nd 7679 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-pnf 10665 df-mnf 10666 df-xr 10667 df-ltxr 10668 df-le 10669 df-ioo 12730 |
This theorem is referenced by: volioofmpt 42156 voliooicof 42158 |
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