Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > smfpreimaltf | 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 |
---|---|
smfpreimaltf.x | ⊢ Ⅎ𝑥𝐹 |
smfpreimaltf.s | ⊢ (𝜑 → 𝑆 ∈ SAlg) |
smfpreimaltf.f | ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) |
smfpreimaltf.d | ⊢ 𝐷 = dom 𝐹 |
smfpreimaltf.a | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
Ref | Expression |
---|---|
smfpreimaltf | ⊢ (𝜑 → {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝐴} ∈ (𝑆 ↾t 𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | smfpreimaltf.a | . 2 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
2 | smfpreimaltf.f | . . . 4 ⊢ (𝜑 → 𝐹 ∈ (SMblFn‘𝑆)) | |
3 | smfpreimaltf.x | . . . . 5 ⊢ Ⅎ𝑥𝐹 | |
4 | smfpreimaltf.s | . . . . 5 ⊢ (𝜑 → 𝑆 ∈ SAlg) | |
5 | smfpreimaltf.d | . . . . 5 ⊢ 𝐷 = dom 𝐹 | |
6 | 3, 4, 5 | issmff 44157 | . . . 4 ⊢ (𝜑 → (𝐹 ∈ (SMblFn‘𝑆) ↔ (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t 𝐷)))) |
7 | 2, 6 | mpbid 231 | . . 3 ⊢ (𝜑 → (𝐷 ⊆ ∪ 𝑆 ∧ 𝐹:𝐷⟶ℝ ∧ ∀𝑎 ∈ ℝ {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t 𝐷))) |
8 | 7 | simp3d 1142 | . 2 ⊢ (𝜑 → ∀𝑎 ∈ ℝ {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t 𝐷)) |
9 | breq2 5074 | . . . . 5 ⊢ (𝑎 = 𝐴 → ((𝐹‘𝑥) < 𝑎 ↔ (𝐹‘𝑥) < 𝐴)) | |
10 | 9 | rabbidv 3404 | . . . 4 ⊢ (𝑎 = 𝐴 → {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑎} = {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝐴}) |
11 | 10 | eleq1d 2823 | . . 3 ⊢ (𝑎 = 𝐴 → ({𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t 𝐷) ↔ {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝐴} ∈ (𝑆 ↾t 𝐷))) |
12 | 11 | rspcva 3550 | . 2 ⊢ ((𝐴 ∈ ℝ ∧ ∀𝑎 ∈ ℝ {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝑎} ∈ (𝑆 ↾t 𝐷)) → {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝐴} ∈ (𝑆 ↾t 𝐷)) |
13 | 1, 8, 12 | syl2anc 583 | 1 ⊢ (𝜑 → {𝑥 ∈ 𝐷 ∣ (𝐹‘𝑥) < 𝐴} ∈ (𝑆 ↾t 𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1085 = wceq 1539 ∈ wcel 2108 Ⅎwnfc 2886 ∀wral 3063 {crab 3067 ⊆ wss 3883 ∪ cuni 4836 class class class wbr 5070 dom cdm 5580 ⟶wf 6414 ‘cfv 6418 (class class class)co 7255 ℝcr 10801 < clt 10940 ↾t crest 17048 SAlgcsalg 43739 SMblFncsmblfn 44123 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-pre-lttri 10876 ax-pre-lttrn 10877 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-op 4565 df-uni 4837 df-iun 4923 df-br 5071 df-opab 5133 df-mpt 5154 df-id 5480 df-po 5494 df-so 5495 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-ov 7258 df-oprab 7259 df-mpo 7260 df-1st 7804 df-2nd 7805 df-er 8456 df-pm 8576 df-en 8692 df-dom 8693 df-sdom 8694 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946 df-ioo 13012 df-ico 13014 df-smblfn 44124 |
This theorem is referenced by: smfpimltmpt 44169 smfpimltxr 44170 |
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