Mathbox for Glauco Siliprandi |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > salpreimagelt | Structured version Visualization version GIF version |
Description: If all the preimages of left-close, 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 2631 | . . . . 5 ⊢ ∪ 𝑆 = 𝐴 |
3 | 2 | a1i 11 | . . . 4 ⊢ (𝜑 → ∪ 𝑆 = 𝐴) |
4 | 3 | difeq1d 3727 | . . 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 10089 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ ℝ*) |
9 | 5, 6, 8 | preimagelt 40912 | . . 3 ⊢ (𝜑 → (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵}) = {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝐶}) |
10 | 4, 9 | eqtr2d 2657 | . 2 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝐶} = (∪ 𝑆 ∖ {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵})) |
11 | salpreimagelt.s | . . 3 ⊢ (𝜑 → 𝑆 ∈ SAlg) | |
12 | id 22 | . . . . 5 ⊢ (𝜑 → 𝜑) | |
13 | 12, 7 | jca 554 | . . . 4 ⊢ (𝜑 → (𝜑 ∧ 𝐶 ∈ ℝ)) |
14 | nfcv 2764 | . . . . 5 ⊢ Ⅎ𝑎𝐶 | |
15 | salpreimagelt.a | . . . . . . 7 ⊢ Ⅎ𝑎𝜑 | |
16 | 14 | nfel1 2779 | . . . . . . 7 ⊢ Ⅎ𝑎 𝐶 ∈ ℝ |
17 | 15, 16 | nfan 1828 | . . . . . 6 ⊢ Ⅎ𝑎(𝜑 ∧ 𝐶 ∈ ℝ) |
18 | nfv 1843 | . . . . . 6 ⊢ Ⅎ𝑎{𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵} ∈ 𝑆 | |
19 | 17, 18 | nfim 1825 | . . . . 5 ⊢ Ⅎ𝑎((𝜑 ∧ 𝐶 ∈ ℝ) → {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵} ∈ 𝑆) |
20 | eleq1 2689 | . . . . . . 7 ⊢ (𝑎 = 𝐶 → (𝑎 ∈ ℝ ↔ 𝐶 ∈ ℝ)) | |
21 | 20 | anbi2d 740 | . . . . . 6 ⊢ (𝑎 = 𝐶 → ((𝜑 ∧ 𝑎 ∈ ℝ) ↔ (𝜑 ∧ 𝐶 ∈ ℝ))) |
22 | breq1 4656 | . . . . . . . 8 ⊢ (𝑎 = 𝐶 → (𝑎 ≤ 𝐵 ↔ 𝐶 ≤ 𝐵)) | |
23 | 22 | rabbidv 3189 | . . . . . . 7 ⊢ (𝑎 = 𝐶 → {𝑥 ∈ 𝐴 ∣ 𝑎 ≤ 𝐵} = {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵}) |
24 | 23 | eleq1d 2686 | . . . . . 6 ⊢ (𝑎 = 𝐶 → ({𝑥 ∈ 𝐴 ∣ 𝑎 ≤ 𝐵} ∈ 𝑆 ↔ {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵} ∈ 𝑆)) |
25 | 21, 24 | imbi12d 334 | . . . . 5 ⊢ (𝑎 = 𝐶 → (((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ 𝐴 ∣ 𝑎 ≤ 𝐵} ∈ 𝑆) ↔ ((𝜑 ∧ 𝐶 ∈ ℝ) → {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵} ∈ 𝑆))) |
26 | salpreimagelt.p | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ 𝐴 ∣ 𝑎 ≤ 𝐵} ∈ 𝑆) | |
27 | 14, 19, 25, 26 | vtoclgf 3264 | . . . 4 ⊢ (𝐶 ∈ ℝ → ((𝜑 ∧ 𝐶 ∈ ℝ) → {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵} ∈ 𝑆)) |
28 | 7, 13, 27 | sylc 65 | . . 3 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵} ∈ 𝑆) |
29 | 11, 28 | saldifcld 40565 | . 2 ⊢ (𝜑 → (∪ 𝑆 ∖ {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵}) ∈ 𝑆) |
30 | 10, 29 | eqeltrd 2701 | 1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝐶} ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 Ⅎwnf 1708 ∈ wcel 1990 {crab 2916 ∖ cdif 3571 ∪ cuni 4436 class class class wbr 4653 ℝcr 9935 ℝ*cxr 10073 < clt 10074 ≤ cle 10075 SAlgcsalg 40528 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-xp 5120 df-cnv 5122 df-xr 10078 df-le 10080 df-salg 40529 |
This theorem is referenced by: salpreimalelt 40938 salpreimagtlt 40939 issmfgelem 40977 |
Copyright terms: Public domain | W3C validator |