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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > smfpreimalt | 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 below is in the subspace sigma-algebra induced by its domain. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
Ref | Expression |
---|---|
smfpreimalt.s | ⊢ (𝜑 → 𝑆 ∈ SAlg) |
smfpreimalt.f | ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) |
smfpreimalt.d | ⊢ 𝐷 = dom 𝐹 |
smfpreimalt.a | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
Ref | Expression |
---|---|
smfpreimalt | ⊢ (𝜑 → {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝐴} ∈ (𝑆 ↾t 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | smfpreimalt.a | . 2 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
2 | smfpreimalt.f | . . . 4 ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) | |
3 | smfpreimalt.s | . . . . 5 ⊢ (𝜑 → 𝑆 ∈ SAlg) | |
4 | smfpreimalt.d | . . . . 5 ⊢ 𝐷 = dom 𝐹 | |
5 | 3, 4 | issmf 43362 | . . . 4 ⊢ (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) ↔ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t 𝐷)))) |
6 | 2, 5 | mpbid 235 | . . 3 ⊢ (𝜑 → (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t 𝐷))) |
7 | 6 | simp3d 1141 | . 2 ⊢ (𝜑 → ∀𝑎 ∈ ℝ {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t 𝐷)) |
8 | breq2 5034 | . . . . 5 ⊢ (𝑎 = 𝐴 → ((𝐹‘𝑥) < 𝑎 ↔ (𝐹‘𝑥) < 𝐴)) | |
9 | 8 | rabbidv 3427 | . . . 4 ⊢ (𝑎 = 𝐴 → {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑎} = {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝐴}) |
10 | 9 | eleq1d 2874 | . . 3 ⊢ (𝑎 = 𝐴 → ({𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t 𝐷) ↔ {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝐴} ∈ (𝑆 ↾t 𝐷))) |
11 | 10 | rspcva 3569 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ ∀𝑎 ∈ ℝ {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t 𝐷)) → {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝐴} ∈ (𝑆 ↾t 𝐷)) |
12 | 1, 7, 11 | syl2anc 587 | 1 ⊢ (𝜑 → {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝐴} ∈ (𝑆 ↾t 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1084 = wceq 1538 ∈ wcel 2111 ∀wral 3106 {crab 3110 ⊆ wss 3881 ∪ cuni 4800 class class class wbr 5030 dom cdm 5519 ⟶wf 6320 ‘cfv 6324 (class class class)co 7135 ℝcr 10525 < clt 10664 ↾t crest 16686 SAlgcsalg 42950 SMblFncsmblfn 43334 |
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 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-pre-lttri 10600 ax-pre-lttrn 10601 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-po 5438 df-so 5439 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-1st 7671 df-2nd 7672 df-er 8272 df-pm 8392 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-ioo 12730 df-ico 12732 df-smblfn 43335 |
This theorem is referenced by: sssmf 43372 smfsssmf 43377 issmfle 43379 smflimlem6 43409 smfco 43434 |
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