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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > smfpimgtxrmptf | Structured version Visualization version GIF version |
Description: Given a function measurable w.r.t. to a sigma-algebra, the preimage of an open interval unbounded above is in the subspace sigma-algebra induced by its domain. (Contributed by Glauco Siliprandi, 20-Dec-2024.) |
Ref | Expression |
---|---|
smfpimgtxrmptf.x | ⊢ Ⅎ𝑥𝜑 |
smfpimgtxrmptf.1 | ⊢ Ⅎ𝑥𝐴 |
smfpimgtxrmptf.s | ⊢ (𝜑 → 𝑆 ∈ SAlg) |
smfpimgtxrmptf.b | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉) |
smfpimgtxrmptf.f | ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ (SMblFn‘𝑆)) |
smfpimgtxrmptf.l | ⊢ (𝜑 → 𝐿 ∈ ℝ*) |
Ref | Expression |
---|---|
smfpimgtxrmptf | ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐿 < 𝐵} ∈ (𝑆 ↾t 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfmpt1 5255 | . . . . . 6 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵) | |
2 | 1 | nfdm 5948 | . . . . 5 ⊢ Ⅎ𝑥dom (𝑥 ∈ 𝐴 ↦ 𝐵) |
3 | nfcv 2903 | . . . . 5 ⊢ Ⅎ𝑦dom (𝑥 ∈ 𝐴 ↦ 𝐵) | |
4 | nfv 1917 | . . . . 5 ⊢ Ⅎ𝑦 𝐿 < ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) | |
5 | nfcv 2903 | . . . . . 6 ⊢ Ⅎ𝑥𝐿 | |
6 | nfcv 2903 | . . . . . 6 ⊢ Ⅎ𝑥 < | |
7 | nfcv 2903 | . . . . . . 7 ⊢ Ⅎ𝑥𝑦 | |
8 | 1, 7 | nffv 6898 | . . . . . 6 ⊢ Ⅎ𝑥((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦) |
9 | 5, 6, 8 | nfbr 5194 | . . . . 5 ⊢ Ⅎ𝑥 𝐿 < ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦) |
10 | fveq2 6888 | . . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦)) | |
11 | 10 | breq2d 5159 | . . . . 5 ⊢ (𝑥 = 𝑦 → (𝐿 < ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ↔ 𝐿 < ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦))) |
12 | 2, 3, 4, 9, 11 | cbvrabw 3467 | . . . 4 ⊢ {𝑥 ∈ dom (𝑥 ∈ 𝐴 ↦ 𝐵) ∣ 𝐿 < ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)} = {𝑦 ∈ dom (𝑥 ∈ 𝐴 ↦ 𝐵) ∣ 𝐿 < ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦)} |
13 | 12 | a1i 11 | . . 3 ⊢ (𝜑 → {𝑥 ∈ dom (𝑥 ∈ 𝐴 ↦ 𝐵) ∣ 𝐿 < ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)} = {𝑦 ∈ dom (𝑥 ∈ 𝐴 ↦ 𝐵) ∣ 𝐿 < ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦)}) |
14 | nfcv 2903 | . . . 4 ⊢ Ⅎ𝑦(𝑥 ∈ 𝐴 ↦ 𝐵) | |
15 | smfpimgtxrmptf.s | . . . 4 ⊢ (𝜑 → 𝑆 ∈ SAlg) | |
16 | smfpimgtxrmptf.f | . . . 4 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ (SMblFn‘𝑆)) | |
17 | eqid 2732 | . . . 4 ⊢ dom (𝑥 ∈ 𝐴 ↦ 𝐵) = dom (𝑥 ∈ 𝐴 ↦ 𝐵) | |
18 | smfpimgtxrmptf.l | . . . 4 ⊢ (𝜑 → 𝐿 ∈ ℝ*) | |
19 | 14, 15, 16, 17, 18 | smfpimgtxr 45482 | . . 3 ⊢ (𝜑 → {𝑦 ∈ dom (𝑥 ∈ 𝐴 ↦ 𝐵) ∣ 𝐿 < ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑦)} ∈ (𝑆 ↾t dom (𝑥 ∈ 𝐴 ↦ 𝐵))) |
20 | 13, 19 | eqeltrd 2833 | . 2 ⊢ (𝜑 → {𝑥 ∈ dom (𝑥 ∈ 𝐴 ↦ 𝐵) ∣ 𝐿 < ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)} ∈ (𝑆 ↾t dom (𝑥 ∈ 𝐴 ↦ 𝐵))) |
21 | smfpimgtxrmptf.x | . . . . . 6 ⊢ Ⅎ𝑥𝜑 | |
22 | smfpimgtxrmptf.1 | . . . . . 6 ⊢ Ⅎ𝑥𝐴 | |
23 | eqid 2732 | . . . . . 6 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
24 | smfpimgtxrmptf.b | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝑉) | |
25 | 21, 22, 23, 24 | dmmptdf2 43920 | . . . . 5 ⊢ (𝜑 → dom (𝑥 ∈ 𝐴 ↦ 𝐵) = 𝐴) |
26 | 2, 22 | rabeqf 3466 | . . . . 5 ⊢ (dom (𝑥 ∈ 𝐴 ↦ 𝐵) = 𝐴 → {𝑥 ∈ dom (𝑥 ∈ 𝐴 ↦ 𝐵) ∣ 𝐿 < ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)} = {𝑥 ∈ 𝐴 ∣ 𝐿 < ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)}) |
27 | 25, 26 | syl 17 | . . . 4 ⊢ (𝜑 → {𝑥 ∈ dom (𝑥 ∈ 𝐴 ↦ 𝐵) ∣ 𝐿 < ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)} = {𝑥 ∈ 𝐴 ∣ 𝐿 < ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)}) |
28 | simpr 485 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴) | |
29 | 22 | fvmpt2f 6996 | . . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝐵 ∈ 𝑉) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = 𝐵) |
30 | 28, 24, 29 | syl2anc 584 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = 𝐵) |
31 | 30 | breq2d 5159 | . . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝐿 < ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) ↔ 𝐿 < 𝐵)) |
32 | 21, 31 | rabbida 3458 | . . . 4 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐿 < ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)} = {𝑥 ∈ 𝐴 ∣ 𝐿 < 𝐵}) |
33 | eqidd 2733 | . . . 4 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐿 < 𝐵} = {𝑥 ∈ 𝐴 ∣ 𝐿 < 𝐵}) | |
34 | 27, 32, 33 | 3eqtrrd 2777 | . . 3 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐿 < 𝐵} = {𝑥 ∈ dom (𝑥 ∈ 𝐴 ↦ 𝐵) ∣ 𝐿 < ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)}) |
35 | 25 | eqcomd 2738 | . . . 4 ⊢ (𝜑 → 𝐴 = dom (𝑥 ∈ 𝐴 ↦ 𝐵)) |
36 | 35 | oveq2d 7421 | . . 3 ⊢ (𝜑 → (𝑆 ↾t 𝐴) = (𝑆 ↾t dom (𝑥 ∈ 𝐴 ↦ 𝐵))) |
37 | 34, 36 | eleq12d 2827 | . 2 ⊢ (𝜑 → ({𝑥 ∈ 𝐴 ∣ 𝐿 < 𝐵} ∈ (𝑆 ↾t 𝐴) ↔ {𝑥 ∈ dom (𝑥 ∈ 𝐴 ↦ 𝐵) ∣ 𝐿 < ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥)} ∈ (𝑆 ↾t dom (𝑥 ∈ 𝐴 ↦ 𝐵)))) |
38 | 20, 37 | mpbird 256 | 1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐿 < 𝐵} ∈ (𝑆 ↾t 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 Ⅎwnf 1785 ∈ wcel 2106 Ⅎwnfc 2883 {crab 3432 class class class wbr 5147 ↦ cmpt 5230 dom cdm 5675 ‘cfv 6540 (class class class)co 7405 ℝ*cxr 11243 < clt 11244 ↾t crest 17362 SAlgcsalg 45010 SMblFncsmblfn 45397 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pow 5362 ax-pr 5426 ax-un 7721 ax-inf2 9632 ax-cc 10426 ax-ac2 10454 ax-cnex 11162 ax-resscn 11163 ax-1cn 11164 ax-icn 11165 ax-addcl 11166 ax-addrcl 11167 ax-mulcl 11168 ax-mulrcl 11169 ax-mulcom 11170 ax-addass 11171 ax-mulass 11172 ax-distr 11173 ax-i2m1 11174 ax-1ne0 11175 ax-1rid 11176 ax-rnegex 11177 ax-rrecex 11178 ax-cnre 11179 ax-pre-lttri 11180 ax-pre-lttrn 11181 ax-pre-ltadd 11182 ax-pre-mulgt0 11183 ax-pre-sup 11184 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rmo 3376 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-int 4950 df-iun 4998 df-iin 4999 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-se 5631 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6297 df-ord 6364 df-on 6365 df-lim 6366 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-isom 6549 df-riota 7361 df-ov 7408 df-oprab 7409 df-mpo 7410 df-om 7852 df-1st 7971 df-2nd 7972 df-frecs 8262 df-wrecs 8293 df-recs 8367 df-rdg 8406 df-1o 8462 df-er 8699 df-map 8818 df-pm 8819 df-en 8936 df-dom 8937 df-sdom 8938 df-fin 8939 df-sup 9433 df-inf 9434 df-card 9930 df-acn 9933 df-ac 10107 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-sub 11442 df-neg 11443 df-div 11868 df-nn 12209 df-n0 12469 df-z 12555 df-uz 12819 df-q 12929 df-rp 12971 df-ioo 13324 df-ico 13326 df-fl 13753 df-rest 17364 df-salg 45011 df-smblfn 45398 |
This theorem is referenced by: smfpimgtxrmpt 45487 smfdmmblpimne 45539 |
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