Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > issmfdmpt | Structured version Visualization version GIF version |
Description: A sufficient condition for "𝐹 being a measurable function w.r.t. to the sigma-algebra 𝑆". (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
Ref | Expression |
---|---|
issmfdmpt.x | ⊢ Ⅎ𝑥𝜑 |
issmfdmpt.a | ⊢ Ⅎ𝑎𝜑 |
issmfdmpt.s | ⊢ (𝜑 → 𝑆 ∈ SAlg) |
issmfdmpt.i | ⊢ (𝜑 → 𝐴 ⊆ ∪ 𝑆) |
issmfdmpt.b | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) |
issmfdmpt.p | ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝑎} ∈ (𝑆 ↾t 𝐴)) |
Ref | Expression |
---|---|
issmfdmpt | ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ (SMblFn‘𝑆)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfmpt1 5156 | . 2 ⊢ Ⅎ𝑥(𝑥 ∈ 𝐴 ↦ 𝐵) | |
2 | issmfdmpt.a | . 2 ⊢ Ⅎ𝑎𝜑 | |
3 | issmfdmpt.s | . 2 ⊢ (𝜑 → 𝑆 ∈ SAlg) | |
4 | issmfdmpt.i | . 2 ⊢ (𝜑 → 𝐴 ⊆ ∪ 𝑆) | |
5 | issmfdmpt.x | . . 3 ⊢ Ⅎ𝑥𝜑 | |
6 | issmfdmpt.b | . . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ) | |
7 | eqid 2821 | . . 3 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
8 | 5, 6, 7 | fmptdf 6875 | . 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶ℝ) |
9 | eqidd 2822 | . . . . . . . . 9 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵)) | |
10 | 9, 6 | fvmpt2d 6775 | . . . . . . . 8 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) = 𝐵) |
11 | 10 | breq1d 5068 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) < 𝑎 ↔ 𝐵 < 𝑎)) |
12 | 11 | ex 415 | . . . . . 6 ⊢ (𝜑 → (𝑥 ∈ 𝐴 → (((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) < 𝑎 ↔ 𝐵 < 𝑎))) |
13 | 5, 12 | ralrimi 3216 | . . . . 5 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 (((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) < 𝑎 ↔ 𝐵 < 𝑎)) |
14 | rabbi 3383 | . . . . 5 ⊢ (∀𝑥 ∈ 𝐴 (((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) < 𝑎 ↔ 𝐵 < 𝑎) ↔ {𝑥 ∈ 𝐴 ∣ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) < 𝑎} = {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝑎}) | |
15 | 13, 14 | sylib 220 | . . . 4 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) < 𝑎} = {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝑎}) |
16 | 15 | adantr 483 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ 𝐴 ∣ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) < 𝑎} = {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝑎}) |
17 | issmfdmpt.p | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝑎} ∈ (𝑆 ↾t 𝐴)) | |
18 | 16, 17 | eqeltrd 2913 | . 2 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ 𝐴 ∣ ((𝑥 ∈ 𝐴 ↦ 𝐵)‘𝑥) < 𝑎} ∈ (𝑆 ↾t 𝐴)) |
19 | 1, 2, 3, 4, 8, 18 | issmfdf 43008 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ (SMblFn‘𝑆)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1533 Ⅎwnf 1780 ∈ wcel 2110 ∀wral 3138 {crab 3142 ⊆ wss 3935 ∪ cuni 4831 class class class wbr 5058 ↦ cmpt 5138 ‘cfv 6349 (class class class)co 7150 ℝcr 10530 < clt 10669 ↾t crest 16688 SAlgcsalg 42587 SMblFncsmblfn 42971 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7455 ax-cnex 10587 ax-resscn 10588 ax-pre-lttri 10605 ax-pre-lttrn 10606 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-pw 4540 df-sn 4561 df-pr 4563 df-op 4567 df-uni 4832 df-iun 4913 df-br 5059 df-opab 5121 df-mpt 5139 df-id 5454 df-po 5468 df-so 5469 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-ov 7153 df-oprab 7154 df-mpo 7155 df-1st 7683 df-2nd 7684 df-er 8283 df-pm 8403 df-en 8504 df-dom 8505 df-sdom 8506 df-pnf 10671 df-mnf 10672 df-xr 10673 df-ltxr 10674 df-le 10675 df-ioo 12736 df-ico 12738 df-smblfn 42972 |
This theorem is referenced by: smfadd 43035 smfrec 43058 smfmul 43064 |
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