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 | ffun 6048 | . . . 4 ⊢ (𝐹:𝐴⟶(ℝ* × ℝ*) → Fun 𝐹) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (𝜑 → Fun 𝐹) |
4 | fvvolioof.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐴) | |
5 | fdm 6051 | . . . . . 6 ⊢ (𝐹:𝐴⟶(ℝ* × ℝ*) → dom 𝐹 = 𝐴) | |
6 | 1, 5 | syl 17 | . . . . 5 ⊢ (𝜑 → dom 𝐹 = 𝐴) |
7 | 6 | eqcomd 2628 | . . . 4 ⊢ (𝜑 → 𝐴 = dom 𝐹) |
8 | 4, 7 | eleqtrd 2703 | . . 3 ⊢ (𝜑 → 𝑋 ∈ dom 𝐹) |
9 | fvco 6274 | . . 3 ⊢ ((Fun 𝐹 ∧ 𝑋 ∈ dom 𝐹) → (((vol ∘ (,)) ∘ 𝐹)‘𝑋) = ((vol ∘ (,))‘(𝐹‘𝑋))) | |
10 | 3, 8, 9 | syl2anc 693 | . 2 ⊢ (𝜑 → (((vol ∘ (,)) ∘ 𝐹)‘𝑋) = ((vol ∘ (,))‘(𝐹‘𝑋))) |
11 | ioof 12271 | . . . . 5 ⊢ (,):(ℝ* × ℝ*)⟶𝒫 ℝ | |
12 | ffun 6048 | . . . . 5 ⊢ ((,):(ℝ* × ℝ*)⟶𝒫 ℝ → Fun (,)) | |
13 | 11, 12 | ax-mp 5 | . . . 4 ⊢ Fun (,) |
14 | 13 | a1i 11 | . . 3 ⊢ (𝜑 → Fun (,)) |
15 | 1, 4 | ffvelrnd 6360 | . . . 4 ⊢ (𝜑 → (𝐹‘𝑋) ∈ (ℝ* × ℝ*)) |
16 | 11 | fdmi 6052 | . . . 4 ⊢ dom (,) = (ℝ* × ℝ*) |
17 | 15, 16 | syl6eleqr 2712 | . . 3 ⊢ (𝜑 → (𝐹‘𝑋) ∈ dom (,)) |
18 | fvco 6274 | . . 3 ⊢ ((Fun (,) ∧ (𝐹‘𝑋) ∈ dom (,)) → ((vol ∘ (,))‘(𝐹‘𝑋)) = (vol‘((,)‘(𝐹‘𝑋)))) | |
19 | 14, 17, 18 | syl2anc 693 | . 2 ⊢ (𝜑 → ((vol ∘ (,))‘(𝐹‘𝑋)) = (vol‘((,)‘(𝐹‘𝑋)))) |
20 | df-ov 6653 | . . . . 5 ⊢ ((1st ‘(𝐹‘𝑋))(,)(2nd ‘(𝐹‘𝑋))) = ((,)‘〈(1st ‘(𝐹‘𝑋)), (2nd ‘(𝐹‘𝑋))〉) | |
21 | 20 | a1i 11 | . . . 4 ⊢ (𝜑 → ((1st ‘(𝐹‘𝑋))(,)(2nd ‘(𝐹‘𝑋))) = ((,)‘〈(1st ‘(𝐹‘𝑋)), (2nd ‘(𝐹‘𝑋))〉)) |
22 | 1st2nd2 7205 | . . . . . . 7 ⊢ ((𝐹‘𝑋) ∈ (ℝ* × ℝ*) → (𝐹‘𝑋) = 〈(1st ‘(𝐹‘𝑋)), (2nd ‘(𝐹‘𝑋))〉) | |
23 | 15, 22 | syl 17 | . . . . . 6 ⊢ (𝜑 → (𝐹‘𝑋) = 〈(1st ‘(𝐹‘𝑋)), (2nd ‘(𝐹‘𝑋))〉) |
24 | 23 | eqcomd 2628 | . . . . 5 ⊢ (𝜑 → 〈(1st ‘(𝐹‘𝑋)), (2nd ‘(𝐹‘𝑋))〉 = (𝐹‘𝑋)) |
25 | 24 | fveq2d 6195 | . . . 4 ⊢ (𝜑 → ((,)‘〈(1st ‘(𝐹‘𝑋)), (2nd ‘(𝐹‘𝑋))〉) = ((,)‘(𝐹‘𝑋))) |
26 | 21, 25 | eqtr2d 2657 | . . 3 ⊢ (𝜑 → ((,)‘(𝐹‘𝑋)) = ((1st ‘(𝐹‘𝑋))(,)(2nd ‘(𝐹‘𝑋)))) |
27 | 26 | fveq2d 6195 | . 2 ⊢ (𝜑 → (vol‘((,)‘(𝐹‘𝑋))) = (vol‘((1st ‘(𝐹‘𝑋))(,)(2nd ‘(𝐹‘𝑋))))) |
28 | 10, 19, 27 | 3eqtrd 2660 | 1 ⊢ (𝜑 → (((vol ∘ (,)) ∘ 𝐹)‘𝑋) = (vol‘((1st ‘(𝐹‘𝑋))(,)(2nd ‘(𝐹‘𝑋))))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 𝒫 cpw 4158 〈cop 4183 × cxp 5112 dom cdm 5114 ∘ ccom 5118 Fun wfun 5882 ⟶wf 5884 ‘cfv 5888 (class class class)co 6650 1st c1st 7166 2nd c2nd 7167 ℝcr 9935 ℝ*cxr 10073 (,)cioo 12175 volcvol 23232 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-pre-lttri 10010 ax-pre-lttrn 10011 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-po 5035 df-so 5036 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-1st 7168 df-2nd 7169 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-ioo 12179 |
This theorem is referenced by: volioofmpt 40211 voliooicof 40213 |
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