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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > salpreimagelt | Structured version Visualization version GIF version |
Description: If all the preimages of left-closed, unbounded below intervals, belong to a sigma-algebra, then all the preimages of right-open, unbounded below intervals, belong to the sigma-algebra. (iv) implies (i) in Proposition 121B of [Fremlin1] p. 36. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
Ref | Expression |
---|---|
salpreimagelt.x | ⊢ Ⅎ𝑥𝜑 |
salpreimagelt.a | ⊢ Ⅎ𝑎𝜑 |
salpreimagelt.s | ⊢ (𝜑 → 𝑆 ∈ SAlg) |
salpreimagelt.u | ⊢ 𝐴 = ∪ 𝑆 |
salpreimagelt.b | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ*) |
salpreimagelt.p | ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ 𝐴 ∣ 𝑎 ≤ 𝐵} ∈ 𝑆) |
salpreimagelt.c | ⊢ (𝜑 → 𝐶 ∈ ℝ) |
Ref | Expression |
---|---|
salpreimagelt | ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝐶} ∈ 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | salpreimagelt.u | . . . . . 6 ⊢ 𝐴 = ∪ 𝑆 | |
2 | 1 | eqcomi 2746 | . . . . 5 ⊢ ∪ 𝑆 = 𝐴 |
3 | 2 | a1i 11 | . . . 4 ⊢ (𝜑 → ∪ 𝑆 = 𝐴) |
4 | 3 | difeq1d 4086 | . . 3 ⊢ (𝜑 → (∪ 𝑆 ∖ {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵}) = (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵})) |
5 | salpreimagelt.x | . . . 4 ⊢ Ⅎ𝑥𝜑 | |
6 | salpreimagelt.b | . . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ*) | |
7 | salpreimagelt.c | . . . . 5 ⊢ (𝜑 → 𝐶 ∈ ℝ) | |
8 | 7 | rexrd 11212 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ ℝ*) |
9 | 5, 6, 8 | preimagelt 45014 | . . 3 ⊢ (𝜑 → (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵}) = {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝐶}) |
10 | 4, 9 | eqtr2d 2778 | . 2 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝐶} = (∪ 𝑆 ∖ {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵})) |
11 | salpreimagelt.s | . . 3 ⊢ (𝜑 → 𝑆 ∈ SAlg) | |
12 | 7 | ancli 550 | . . . 4 ⊢ (𝜑 → (𝜑 ∧ 𝐶 ∈ ℝ)) |
13 | salpreimagelt.a | . . . . . . 7 ⊢ Ⅎ𝑎𝜑 | |
14 | nfcv 2908 | . . . . . . . 8 ⊢ Ⅎ𝑎𝐶 | |
15 | 14 | nfel1 2924 | . . . . . . 7 ⊢ Ⅎ𝑎 𝐶 ∈ ℝ |
16 | 13, 15 | nfan 1903 | . . . . . 6 ⊢ Ⅎ𝑎(𝜑 ∧ 𝐶 ∈ ℝ) |
17 | nfv 1918 | . . . . . 6 ⊢ Ⅎ𝑎{𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵} ∈ 𝑆 | |
18 | 16, 17 | nfim 1900 | . . . . 5 ⊢ Ⅎ𝑎((𝜑 ∧ 𝐶 ∈ ℝ) → {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵} ∈ 𝑆) |
19 | eleq1 2826 | . . . . . . 7 ⊢ (𝑎 = 𝐶 → (𝑎 ∈ ℝ ↔ 𝐶 ∈ ℝ)) | |
20 | 19 | anbi2d 630 | . . . . . 6 ⊢ (𝑎 = 𝐶 → ((𝜑 ∧ 𝑎 ∈ ℝ) ↔ (𝜑 ∧ 𝐶 ∈ ℝ))) |
21 | breq1 5113 | . . . . . . . 8 ⊢ (𝑎 = 𝐶 → (𝑎 ≤ 𝐵 ↔ 𝐶 ≤ 𝐵)) | |
22 | 21 | rabbidv 3418 | . . . . . . 7 ⊢ (𝑎 = 𝐶 → {𝑥 ∈ 𝐴 ∣ 𝑎 ≤ 𝐵} = {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵}) |
23 | 22 | eleq1d 2823 | . . . . . 6 ⊢ (𝑎 = 𝐶 → ({𝑥 ∈ 𝐴 ∣ 𝑎 ≤ 𝐵} ∈ 𝑆 ↔ {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵} ∈ 𝑆)) |
24 | 20, 23 | imbi12d 345 | . . . . 5 ⊢ (𝑎 = 𝐶 → (((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ 𝐴 ∣ 𝑎 ≤ 𝐵} ∈ 𝑆) ↔ ((𝜑 ∧ 𝐶 ∈ ℝ) → {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵} ∈ 𝑆))) |
25 | salpreimagelt.p | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ 𝐴 ∣ 𝑎 ≤ 𝐵} ∈ 𝑆) | |
26 | 18, 24, 25 | vtoclg1f 3527 | . . . 4 ⊢ (𝐶 ∈ ℝ → ((𝜑 ∧ 𝐶 ∈ ℝ) → {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵} ∈ 𝑆)) |
27 | 7, 12, 26 | sylc 65 | . . 3 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵} ∈ 𝑆) |
28 | 11, 27 | saldifcld 44662 | . 2 ⊢ (𝜑 → (∪ 𝑆 ∖ {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵}) ∈ 𝑆) |
29 | 10, 28 | eqeltrd 2838 | 1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝐶} ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 Ⅎwnf 1786 ∈ wcel 2107 {crab 3410 ∖ cdif 3912 ∪ cuni 4870 class class class wbr 5110 ℝcr 11057 ℝ*cxr 11195 < clt 11196 ≤ cle 11197 SAlgcsalg 44623 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pr 5389 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-nfc 2890 df-ral 3066 df-rex 3075 df-rab 3411 df-v 3450 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-br 5111 df-opab 5173 df-xp 5644 df-cnv 5646 df-xr 11200 df-le 11202 df-salg 44624 |
This theorem is referenced by: salpreimalelt 45044 salpreimagtlt 45045 issmfgelem 45084 |
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