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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > issmf | Structured version Visualization version GIF version |
Description: The predicate "𝐹 is a real-valued measurable function w.r.t. to the sigma-algebra 𝑆". A function is measurable iff the preimages of all open intervals unbounded below are in the subspace sigma-algebra induced by its domain. The domain of 𝐹 is required to be a subset of the underlying set of 𝑆. Definition 121C of [Fremlin1] p. 36, and Proposition 121B (i) of [Fremlin1] p. 35 . (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
Ref | Expression |
---|---|
issmf.s | ⊢ (𝜑 → 𝑆 ∈ SAlg) |
issmf.d | ⊢ 𝐷 = dom 𝐹 |
Ref | Expression |
---|---|
issmf | ⊢ (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) ↔ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t 𝐷)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | issmf.s | . . 3 ⊢ (𝜑 → 𝑆 ∈ SAlg) | |
2 | issmf.d | . . 3 ⊢ 𝐷 = dom 𝐹 | |
3 | 1, 2 | issmflem 46683 | . 2 ⊢ (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) ↔ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) < 𝑏} ∈ (𝑆 ↾t 𝐷)))) |
4 | breq2 5152 | . . . . . . . 8 ⊢ (𝑏 = 𝑎 → ((𝐹‘𝑦) < 𝑏 ↔ (𝐹‘𝑦) < 𝑎)) | |
5 | 4 | rabbidv 3441 | . . . . . . 7 ⊢ (𝑏 = 𝑎 → {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) < 𝑏} = {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) < 𝑎}) |
6 | 5 | eleq1d 2824 | . . . . . 6 ⊢ (𝑏 = 𝑎 → ({𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) < 𝑏} ∈ (𝑆 ↾t 𝐷) ↔ {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) < 𝑎} ∈ (𝑆 ↾t 𝐷))) |
7 | fveq2 6907 | . . . . . . . . . 10 ⊢ (𝑦 = 𝑥 → (𝐹‘𝑦) = (𝐹‘𝑥)) | |
8 | 7 | breq1d 5158 | . . . . . . . . 9 ⊢ (𝑦 = 𝑥 → ((𝐹‘𝑦) < 𝑎 ↔ (𝐹‘𝑥) < 𝑎)) |
9 | 8 | cbvrabv 3444 | . . . . . . . 8 ⊢ {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) < 𝑎} = {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑎} |
10 | 9 | eleq1i 2830 | . . . . . . 7 ⊢ ({𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) < 𝑎} ∈ (𝑆 ↾t 𝐷) ↔ {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t 𝐷)) |
11 | 10 | a1i 11 | . . . . . 6 ⊢ (𝑏 = 𝑎 → ({𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) < 𝑎} ∈ (𝑆 ↾t 𝐷) ↔ {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t 𝐷))) |
12 | 6, 11 | bitrd 279 | . . . . 5 ⊢ (𝑏 = 𝑎 → ({𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) < 𝑏} ∈ (𝑆 ↾t 𝐷) ↔ {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t 𝐷))) |
13 | 12 | cbvralvw 3235 | . . . 4 ⊢ (∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) < 𝑏} ∈ (𝑆 ↾t 𝐷) ↔ ∀𝑎 ∈ ℝ {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t 𝐷)) |
14 | 13 | 3anbi3i 1158 | . . 3 ⊢ ((𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) < 𝑏} ∈ (𝑆 ↾t 𝐷)) ↔ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t 𝐷))) |
15 | 14 | a1i 11 | . 2 ⊢ (𝜑 → ((𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑏 ∈ ℝ {𝑦 ∈ 𝐷 ∣ (𝐹‘𝑦) < 𝑏} ∈ (𝑆 ↾t 𝐷)) ↔ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t 𝐷)))) |
16 | 3, 15 | bitrd 279 | 1 ⊢ (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) ↔ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t 𝐷)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ∀wral 3059 {crab 3433 ⊆ wss 3963 ∪ cuni 4912 class class class wbr 5148 dom cdm 5689 ⟶wf 6559 ‘cfv 6563 (class class class)co 7431 ℝcr 11152 < clt 11293 ↾t crest 17467 SAlgcsalg 46264 SMblFncsmblfn 46651 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-pre-lttri 11227 ax-pre-lttrn 11228 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-nel 3045 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-po 5597 df-so 5598 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8013 df-2nd 8014 df-er 8744 df-pm 8868 df-en 8985 df-dom 8986 df-sdom 8987 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-ioo 13388 df-ico 13390 df-smblfn 46652 |
This theorem is referenced by: smfpreimalt 46687 smff 46688 smfdmss 46689 issmff 46690 issmfd 46691 issmflelem 46700 issmfgtlem 46711 issmfgelem 46725 |
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