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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 6258 | . . 3 ⊢ (𝜑 → Fun 𝐹) |
3 | fvvolioof.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐴) | |
4 | 1 | fdmd 6263 | . . . . 5 ⊢ (𝜑 → dom 𝐹 = 𝐴) |
5 | 4 | eqcomd 2803 | . . . 4 ⊢ (𝜑 → 𝐴 = dom 𝐹) |
6 | 3, 5 | eleqtrd 2878 | . . 3 ⊢ (𝜑 → 𝑋 ∈ dom 𝐹) |
7 | fvco 6497 | . . 3 ⊢ ((Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹) → (((vol ∘ (,)) ∘ 𝐹)‘𝑋) = ((vol ∘ (,))‘(𝐹‘𝑋))) | |
8 | 2, 6, 7 | syl2anc 580 | . 2 ⊢ (𝜑 → (((vol ∘ (,)) ∘ 𝐹)‘𝑋) = ((vol ∘ (,))‘(𝐹‘𝑋))) |
9 | ioof 12517 | . . . . 5 ⊢ (,):(ℝ* × ℝ*)⟶𝒫 ℝ | |
10 | ffun 6257 | . . . . 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 6584 | . . . 4 ⊢ (𝜑 → (𝐹‘𝑋) ∈ (ℝ* × ℝ*)) |
14 | 9 | fdmi 6264 | . . . 4 ⊢ dom (,) = (ℝ* × ℝ*) |
15 | 13, 14 | syl6eleqr 2887 | . . 3 ⊢ (𝜑 → (𝐹‘𝑋) ∈ dom (,)) |
16 | fvco 6497 | . . 3 ⊢ ((Fun (,) ∧ (𝐹‘𝑋) ∈ dom (,)) → ((vol ∘ (,))‘(𝐹‘𝑋)) = (vol‘((,)‘(𝐹‘𝑋)))) | |
17 | 12, 15, 16 | syl2anc 580 | . 2 ⊢ (𝜑 → ((vol ∘ (,))‘(𝐹‘𝑋)) = (vol‘((,)‘(𝐹‘𝑋)))) |
18 | df-ov 6879 | . . . . 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 7438 | . . . . . . 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 2803 | . . . . 5 ⊢ (𝜑 → 〈(1st ‘(𝐹‘𝑋)), (2nd ‘(𝐹‘𝑋))〉 = (𝐹‘𝑋)) |
23 | 22 | fveq2d 6413 | . . . 4 ⊢ (𝜑 → ((,)‘〈(1st ‘(𝐹‘𝑋)), (2nd ‘(𝐹‘𝑋))〉) = ((,)‘(𝐹‘𝑋))) |
24 | 19, 23 | eqtr2d 2832 | . . 3 ⊢ (𝜑 → ((,)‘(𝐹‘𝑋)) = ((1st ‘(𝐹‘𝑋))(,)(2nd ‘(𝐹‘𝑋)))) |
25 | 24 | fveq2d 6413 | . 2 ⊢ (𝜑 → (vol‘((,)‘(𝐹‘𝑋))) = (vol‘((1st ‘(𝐹‘𝑋))(,)(2nd ‘(𝐹‘𝑋))))) |
26 | 8, 17, 25 | 3eqtrd 2835 | 1 ⊢ (𝜑 → (((vol ∘ (,)) ∘ 𝐹)‘𝑋) = (vol‘((1st ‘(𝐹‘𝑋))(,)(2nd ‘(𝐹‘𝑋))))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1653 ∈ wcel 2157 𝒫 cpw 4347 〈cop 4372 × cxp 5308 dom cdm 5310 ∘ ccom 5314 Fun wfun 6093 ⟶wf 6095 ‘cfv 6099 (class class class)co 6876 1st c1st 7397 2nd c2nd 7398 ℝcr 10221 ℝ*cxr 10360 (,)cioo 12420 volcvol 23568 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1891 ax-4 1905 ax-5 2006 ax-6 2072 ax-7 2107 ax-8 2159 ax-9 2166 ax-10 2185 ax-11 2200 ax-12 2213 ax-13 2354 ax-ext 2775 ax-sep 4973 ax-nul 4981 ax-pow 5033 ax-pr 5095 ax-un 7181 ax-cnex 10278 ax-resscn 10279 ax-pre-lttri 10296 ax-pre-lttrn 10297 |
This theorem depends on definitions: df-bi 199 df-an 386 df-or 875 df-3or 1109 df-3an 1110 df-tru 1657 df-ex 1876 df-nf 1880 df-sb 2065 df-mo 2590 df-eu 2607 df-clab 2784 df-cleq 2790 df-clel 2793 df-nfc 2928 df-ne 2970 df-nel 3073 df-ral 3092 df-rex 3093 df-rab 3096 df-v 3385 df-sbc 3632 df-csb 3727 df-dif 3770 df-un 3772 df-in 3774 df-ss 3781 df-nul 4114 df-if 4276 df-pw 4349 df-sn 4367 df-pr 4369 df-op 4373 df-uni 4627 df-iun 4710 df-br 4842 df-opab 4904 df-mpt 4921 df-id 5218 df-po 5231 df-so 5232 df-xp 5316 df-rel 5317 df-cnv 5318 df-co 5319 df-dm 5320 df-rn 5321 df-res 5322 df-ima 5323 df-iota 6062 df-fun 6101 df-fn 6102 df-f 6103 df-f1 6104 df-fo 6105 df-f1o 6106 df-fv 6107 df-ov 6879 df-oprab 6880 df-mpt2 6881 df-1st 7399 df-2nd 7400 df-er 7980 df-en 8194 df-dom 8195 df-sdom 8196 df-pnf 10363 df-mnf 10364 df-xr 10365 df-ltxr 10366 df-le 10367 df-ioo 12424 |
This theorem is referenced by: volioofmpt 40942 voliooicof 40944 |
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