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| Mirrors > Home > MPE Home > Th. List > Mathboxes > salpreimagtlt | Structured version Visualization version GIF version | ||
| Description: If all the preimages of lef-open, unbounded above intervals, belong to a sigma-algebra, then all the preimages of right-open, unbounded below intervals, belong to the sigma-algebra. (iii) implies (i) in Proposition 121B of [Fremlin1] p. 36. (Contributed by Glauco Siliprandi, 26-Jun-2021.) | 
| Ref | Expression | 
|---|---|
| salpreimagtlt.x | ⊢ Ⅎ𝑥𝜑 | 
| salpreimagtlt.a | ⊢ Ⅎ𝑎𝜑 | 
| salpreimagtlt.s | ⊢ (𝜑 → 𝑆 ∈ SAlg) | 
| salpreimagtlt.u | ⊢ 𝐴 = ∪ 𝑆 | 
| salpreimagtlt.b | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ*) | 
| salpreimagtlt.p | ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ 𝐴 ∣ 𝑎 < 𝐵} ∈ 𝑆) | 
| salpreimagtlt.c | ⊢ (𝜑 → 𝐶 ∈ ℝ) | 
| Ref | Expression | 
|---|---|
| salpreimagtlt | ⊢ (𝜑 → {𝑥 ∈ 𝐴 ∣ 𝐵 < 𝐶} ∈ 𝑆) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | salpreimagtlt.x | . 2 ⊢ Ⅎ𝑥𝜑 | |
| 2 | salpreimagtlt.a | . 2 ⊢ Ⅎ𝑎𝜑 | |
| 3 | salpreimagtlt.s | . 2 ⊢ (𝜑 → 𝑆 ∈ SAlg) | |
| 4 | salpreimagtlt.u | . 2 ⊢ 𝐴 = ∪ 𝑆 | |
| 5 | salpreimagtlt.b | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ*) | |
| 6 | nfv 1913 | . . . 4 ⊢ Ⅎ𝑥 𝑎 ∈ ℝ | |
| 7 | 1, 6 | nfan 1898 | . . 3 ⊢ Ⅎ𝑥(𝜑 ∧ 𝑎 ∈ ℝ) | 
| 8 | nfv 1913 | . . 3 ⊢ Ⅎ𝑏(𝜑 ∧ 𝑎 ∈ ℝ) | |
| 9 | 3 | adantr 480 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → 𝑆 ∈ SAlg) | 
| 10 | 5 | adantlr 715 | . . 3 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ ℝ*) | 
| 11 | nfv 1913 | . . . . . . 7 ⊢ Ⅎ𝑎 𝑏 ∈ ℝ | |
| 12 | 2, 11 | nfan 1898 | . . . . . 6 ⊢ Ⅎ𝑎(𝜑 ∧ 𝑏 ∈ ℝ) | 
| 13 | nfv 1913 | . . . . . 6 ⊢ Ⅎ𝑎{𝑥 ∈ 𝐴 ∣ 𝑏 < 𝐵} ∈ 𝑆 | |
| 14 | 12, 13 | nfim 1895 | . . . . 5 ⊢ Ⅎ𝑎((𝜑 ∧ 𝑏 ∈ ℝ) → {𝑥 ∈ 𝐴 ∣ 𝑏 < 𝐵} ∈ 𝑆) | 
| 15 | eleq1w 2823 | . . . . . . 7 ⊢ (𝑎 = 𝑏 → (𝑎 ∈ ℝ ↔ 𝑏 ∈ ℝ)) | |
| 16 | 15 | anbi2d 630 | . . . . . 6 ⊢ (𝑎 = 𝑏 → ((𝜑 ∧ 𝑎 ∈ ℝ) ↔ (𝜑 ∧ 𝑏 ∈ ℝ))) | 
| 17 | breq1 5145 | . . . . . . . 8 ⊢ (𝑎 = 𝑏 → (𝑎 < 𝐵 ↔ 𝑏 < 𝐵)) | |
| 18 | 17 | rabbidv 3443 | . . . . . . 7 ⊢ (𝑎 = 𝑏 → {𝑥 ∈ 𝐴 ∣ 𝑎 < 𝐵} = {𝑥 ∈ 𝐴 ∣ 𝑏 < 𝐵}) | 
| 19 | 18 | eleq1d 2825 | . . . . . 6 ⊢ (𝑎 = 𝑏 → ({𝑥 ∈ 𝐴 ∣ 𝑎 < 𝐵} ∈ 𝑆 ↔ {𝑥 ∈ 𝐴 ∣ 𝑏 < 𝐵} ∈ 𝑆)) | 
| 20 | 16, 19 | imbi12d 344 | . . . . 5 ⊢ (𝑎 = 𝑏 → (((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ 𝐴 ∣ 𝑎 < 𝐵} ∈ 𝑆) ↔ ((𝜑 ∧ 𝑏 ∈ ℝ) → {𝑥 ∈ 𝐴 ∣ 𝑏 < 𝐵} ∈ 𝑆))) | 
| 21 | salpreimagtlt.p | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ 𝐴 ∣ 𝑎 < 𝐵} ∈ 𝑆) | |
| 22 | 14, 20, 21 | chvarfv 2239 | . . . 4 ⊢ ((𝜑 ∧ 𝑏 ∈ ℝ) → {𝑥 ∈ 𝐴 ∣ 𝑏 < 𝐵} ∈ 𝑆) | 
| 23 | 22 | adantlr 715 | . . 3 ⊢ (((𝜑 ∧ 𝑎 ∈ ℝ) ∧ 𝑏 ∈ ℝ) → {𝑥 ∈ 𝐴 ∣ 𝑏 < 𝐵} ∈ 𝑆) | 
| 24 | simpr 484 | . . 3 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → 𝑎 ∈ ℝ) | |
| 25 | 7, 8, 9, 10, 23, 24 | salpreimagtge 46745 | . 2 ⊢ ((𝜑 ∧ 𝑎 ∈ ℝ) → {𝑥 ∈ 𝐴 ∣ 𝑎 ≤ 𝐵} ∈ 𝑆) | 
| 26 | salpreimagtlt.c | . 2 ⊢ (𝜑 → 𝐶 ∈ ℝ) | |
| 27 | 1, 2, 3, 4, 5, 25, 26 | salpreimagelt 46727 | 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 ∪ cuni 4906 class class class wbr 5142 ℝcr 11155 ℝ*cxr 11295 < clt 11296 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-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-inf2 9682 ax-cnex 11212 ax-resscn 11213 ax-1cn 11214 ax-icn 11215 ax-addcl 11216 ax-addrcl 11217 ax-mulcl 11218 ax-mulrcl 11219 ax-mulcom 11220 ax-addass 11221 ax-mulass 11222 ax-distr 11223 ax-i2m1 11224 ax-1ne0 11225 ax-1rid 11226 ax-rnegex 11227 ax-rrecex 11228 ax-cnre 11229 ax-pre-lttri 11230 ax-pre-lttrn 11231 ax-pre-ltadd 11232 ax-pre-mulgt0 11233 ax-pre-sup 11234 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-nel 3046 df-ral 3061 df-rex 3070 df-rmo 3379 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-int 4946 df-iun 4992 df-iin 4993 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-se 5637 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-isom 6569 df-riota 7389 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-1st 8015 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-er 8746 df-map 8869 df-en 8987 df-dom 8988 df-sdom 8989 df-sup 9483 df-inf 9484 df-card 9980 df-acn 9983 df-pnf 11298 df-mnf 11299 df-xr 11300 df-ltxr 11301 df-le 11302 df-sub 11495 df-neg 11496 df-div 11922 df-nn 12268 df-n0 12529 df-z 12616 df-uz 12880 df-q 12992 df-rp 13036 df-fl 13833 df-salg 46329 | 
| This theorem is referenced by: issmfgtlem 46775 | 
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