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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 6727 | . . 3 ⊢ (𝜑 → Fun 𝐹) |
3 | fvvolioof.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐴) | |
4 | 1 | fdmd 6733 | . . . . 5 ⊢ (𝜑 → dom 𝐹 = 𝐴) |
5 | 4 | eqcomd 2731 | . . . 4 ⊢ (𝜑 → 𝐴 = dom 𝐹) |
6 | 3, 5 | eleqtrd 2827 | . . 3 ⊢ (𝜑 → 𝑋 ∈ dom 𝐹) |
7 | fvco 6995 | . . 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 | ioof 13459 | . . . . 5 ⊢ (,):(ℝ* × ℝ*)⟶𝒫 ℝ | |
10 | ffun 6726 | . . . . 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 7094 | . . . 4 ⊢ (𝜑 → (𝐹‘𝑋) ∈ (ℝ* × ℝ*)) |
14 | 9 | fdmi 6734 | . . . 4 ⊢ dom (,) = (ℝ* × ℝ*) |
15 | 13, 14 | eleqtrrdi 2836 | . . 3 ⊢ (𝜑 → (𝐹‘𝑋) ∈ dom (,)) |
16 | fvco 6995 | . . 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 7422 | . . . . 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 8033 | . . . . . . 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 2731 | . . . . 5 ⊢ (𝜑 → 〈(1st ‘(𝐹‘𝑋)), (2nd ‘(𝐹‘𝑋))〉 = (𝐹‘𝑋)) |
23 | 22 | fveq2d 6900 | . . . 4 ⊢ (𝜑 → ((,)‘〈(1st ‘(𝐹‘𝑋)), (2nd ‘(𝐹‘𝑋))〉) = ((,)‘(𝐹‘𝑋))) |
24 | 19, 23 | eqtr2d 2766 | . . 3 ⊢ (𝜑 → ((,)‘(𝐹‘𝑋)) = ((1st ‘(𝐹‘𝑋))(,)(2nd ‘(𝐹‘𝑋)))) |
25 | 24 | fveq2d 6900 | . 2 ⊢ (𝜑 → (vol‘((,)‘(𝐹‘𝑋))) = (vol‘((1st ‘(𝐹‘𝑋))(,)(2nd ‘(𝐹‘𝑋))))) |
26 | 8, 17, 25 | 3eqtrd 2769 | 1 ⊢ (𝜑 → (((vol ∘ (,)) ∘ 𝐹)‘𝑋) = (vol‘((1st ‘(𝐹‘𝑋))(,)(2nd ‘(𝐹‘𝑋))))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2098 𝒫 cpw 4604 〈cop 4636 × cxp 5676 dom cdm 5678 ∘ ccom 5682 Fun wfun 6543 ⟶wf 6545 ‘cfv 6549 (class class class)co 7419 1st c1st 7992 2nd c2nd 7993 ℝcr 11139 ℝ*cxr 11279 (,)cioo 13359 volcvol 25436 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2696 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 ax-cnex 11196 ax-resscn 11197 ax-pre-lttri 11214 ax-pre-lttrn 11215 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-nel 3036 df-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5576 df-po 5590 df-so 5591 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 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-ov 7422 df-oprab 7423 df-mpo 7424 df-1st 7994 df-2nd 7995 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11282 df-mnf 11283 df-xr 11284 df-ltxr 11285 df-le 11286 df-ioo 13363 |
This theorem is referenced by: volioofmpt 45520 voliooicof 45522 |
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