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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 2745 | . . . . 5 ⊢ ∪ 𝑆 = 𝐴 | 
| 3 | 2 | a1i 11 | . . . 4 ⊢ (𝜑 → ∪ 𝑆 = 𝐴) | 
| 4 | 3 | difeq1d 4124 | . . 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 11312 | . . . 4 ⊢ (𝜑 → 𝐶 ∈ ℝ*) | 
| 9 | 5, 6, 8 | preimagelt 46719 | . . 3 ⊢ (𝜑 → (𝐴 ∖ {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵}) = {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝐶}) | 
| 10 | 4, 9 | eqtr2d 2777 | . 2 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝐶} = (∪ 𝑆 ∖ {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵})) | 
| 11 | salpreimagelt.s | . . 3 ⊢ (𝜑 → 𝑆 ∈ SAlg) | |
| 12 | 7 | ancli 548 | . . . 4 ⊢ (𝜑 → (𝜑 ∧ 𝐶 ∈ ℝ)) | 
| 13 | salpreimagelt.a | . . . . . . 7 ⊢ Ⅎ𝑎𝜑 | |
| 14 | nfcv 2904 | . . . . . . . 8 ⊢ Ⅎ𝑎𝐶 | |
| 15 | 14 | nfel1 2921 | . . . . . . 7 ⊢ Ⅎ𝑎 𝐶 ∈ ℝ | 
| 16 | 13, 15 | nfan 1898 | . . . . . 6 ⊢ Ⅎ𝑎(𝜑 ∧ 𝐶 ∈ ℝ) | 
| 17 | nfv 1913 | . . . . . 6 ⊢ Ⅎ𝑎{𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵} ∈ 𝑆 | |
| 18 | 16, 17 | nfim 1895 | . . . . 5 ⊢ Ⅎ𝑎((𝜑 ∧ 𝐶 ∈ ℝ) → {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵} ∈ 𝑆) | 
| 19 | eleq1 2828 | . . . . . . 7 ⊢ (𝑎 = 𝐶 → (𝑎 ∈ ℝ ↔ 𝐶 ∈ ℝ)) | |
| 20 | 19 | anbi2d 630 | . . . . . 6 ⊢ (𝑎 = 𝐶 → ((𝜑 ∧ 𝑎 ∈ ℝ) ↔ (𝜑 ∧ 𝐶 ∈ ℝ))) | 
| 21 | breq1 5145 | . . . . . . . 8 ⊢ (𝑎 = 𝐶 → (𝑎 ≤ 𝐵 ↔ 𝐶 ≤ 𝐵)) | |
| 22 | 21 | rabbidv 3443 | . . . . . . 7 ⊢ (𝑎 = 𝐶 → {𝑥 ∈ 𝐴 ∣ 𝑎 ≤ 𝐵} = {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵}) | 
| 23 | 22 | eleq1d 2825 | . . . . . 6 ⊢ (𝑎 = 𝐶 → ({𝑥 ∈ 𝐴 ∣ 𝑎 ≤ 𝐵} ∈ 𝑆 ↔ {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵} ∈ 𝑆)) | 
| 24 | 20, 23 | imbi12d 344 | . . . . 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 3569 | . . . 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 46367 | . 2 ⊢ (𝜑 → (∪ 𝑆 ∖ {𝑥 ∈ 𝐴 ∣ 𝐶 ≤ 𝐵}) ∈ 𝑆) | 
| 29 | 10, 28 | eqeltrd 2840 | 1 ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝐶} ∈ 𝑆) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1539 Ⅎwnf 1782 ∈ wcel 2107 {crab 3435 ∖ cdif 3947 ∪ cuni 4906 class class class wbr 5142 ℝcr 11155 ℝ*cxr 11295 < clt 11296 ≤ cle 11297 SAlgcsalg 46328 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-xp 5690 df-cnv 5692 df-xr 11300 df-le 11302 df-salg 46329 | 
| This theorem is referenced by: salpreimalelt 46749 salpreimagtlt 46750 issmfgelem 46789 | 
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