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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > saliunclf | Structured version Visualization version GIF version |
Description: SAlg sigma-algebra is closed under countable indexed union. (Contributed by Glauco Siliprandi, 24-Jan-2025.) |
Ref | Expression |
---|---|
saliunclf.1 | ⊢ Ⅎ𝑘𝜑 |
saliunclf.2 | ⊢ Ⅎ𝑘𝑆 |
saliunclf.3 | ⊢ Ⅎ𝑘𝐾 |
saliunclf.4 | ⊢ (𝜑 → 𝑆 ∈ SAlg) |
saliunclf.5 | ⊢ (𝜑 → 𝐾 ≼ ω) |
saliunclf.6 | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐾) → 𝐸 ∈ 𝑆) |
Ref | Expression |
---|---|
saliunclf | ⊢ (𝜑 → ∪ 𝑘 ∈ 𝐾 𝐸 ∈ 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | saliunclf.1 | . . . 4 ⊢ Ⅎ𝑘𝜑 | |
2 | saliunclf.6 | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐾) → 𝐸 ∈ 𝑆) | |
3 | 1, 2 | ralrimia 3252 | . . 3 ⊢ (𝜑 → ∀𝑘 ∈ 𝐾 𝐸 ∈ 𝑆) |
4 | dfiun3g 5939 | . . 3 ⊢ (∀𝑘 ∈ 𝐾 𝐸 ∈ 𝑆 → ∪ 𝑘 ∈ 𝐾 𝐸 = ∪ ran (𝑘 ∈ 𝐾 ↦ 𝐸)) | |
5 | 3, 4 | syl 17 | . 2 ⊢ (𝜑 → ∪ 𝑘 ∈ 𝐾 𝐸 = ∪ ran (𝑘 ∈ 𝐾 ↦ 𝐸)) |
6 | saliunclf.4 | . . 3 ⊢ (𝜑 → 𝑆 ∈ SAlg) | |
7 | saliunclf.3 | . . . . 5 ⊢ Ⅎ𝑘𝐾 | |
8 | saliunclf.2 | . . . . 5 ⊢ Ⅎ𝑘𝑆 | |
9 | eqid 2731 | . . . . 5 ⊢ (𝑘 ∈ 𝐾 ↦ 𝐸) = (𝑘 ∈ 𝐾 ↦ 𝐸) | |
10 | 1, 7, 8, 9, 2 | rnmptssdff 43658 | . . . 4 ⊢ (𝜑 → ran (𝑘 ∈ 𝐾 ↦ 𝐸) ⊆ 𝑆) |
11 | 6, 10 | sselpwd 5303 | . . 3 ⊢ (𝜑 → ran (𝑘 ∈ 𝐾 ↦ 𝐸) ∈ 𝒫 𝑆) |
12 | saliunclf.5 | . . . 4 ⊢ (𝜑 → 𝐾 ≼ ω) | |
13 | 7 | rn1st 43656 | . . . 4 ⊢ (𝐾 ≼ ω → ran (𝑘 ∈ 𝐾 ↦ 𝐸) ≼ ω) |
14 | 12, 13 | syl 17 | . . 3 ⊢ (𝜑 → ran (𝑘 ∈ 𝐾 ↦ 𝐸) ≼ ω) |
15 | 6, 11, 14 | salunicl 44710 | . 2 ⊢ (𝜑 → ∪ ran (𝑘 ∈ 𝐾 ↦ 𝐸) ∈ 𝑆) |
16 | 5, 15 | eqeltrd 2832 | 1 ⊢ (𝜑 → ∪ 𝑘 ∈ 𝐾 𝐸 ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1541 Ⅎwnf 1785 ∈ wcel 2106 Ⅎwnfc 2882 ∀wral 3060 ∪ cuni 4885 ∪ ciun 4974 class class class wbr 5125 ↦ cmpt 5208 ran crn 5654 ωcom 7822 ≼ cdom 8903 SAlgcsalg 44702 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-rep 5262 ax-sep 5276 ax-nul 5283 ax-pow 5340 ax-pr 5404 ax-un 7692 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rmo 3364 df-reu 3365 df-rab 3419 df-v 3461 df-sbc 3758 df-csb 3874 df-dif 3931 df-un 3933 df-in 3935 df-ss 3945 df-pss 3947 df-nul 4303 df-if 4507 df-pw 4582 df-sn 4607 df-pr 4609 df-op 4613 df-uni 4886 df-int 4928 df-iun 4976 df-br 5126 df-opab 5188 df-mpt 5209 df-tr 5243 df-id 5551 df-eprel 5557 df-po 5565 df-so 5566 df-fr 5608 df-se 5609 df-we 5610 df-xp 5659 df-rel 5660 df-cnv 5661 df-co 5662 df-dm 5663 df-rn 5664 df-res 5665 df-ima 5666 df-pred 6273 df-ord 6340 df-on 6341 df-lim 6342 df-suc 6343 df-iota 6468 df-fun 6518 df-fn 6519 df-f 6520 df-f1 6521 df-fo 6522 df-f1o 6523 df-fv 6524 df-isom 6525 df-riota 7333 df-ov 7380 df-oprab 7381 df-mpo 7382 df-om 7823 df-1st 7941 df-2nd 7942 df-frecs 8232 df-wrecs 8263 df-recs 8337 df-er 8670 df-map 8789 df-en 8906 df-dom 8907 df-card 9899 df-acn 9902 df-salg 44703 |
This theorem is referenced by: saliuncl 44717 saliinclf 44720 smfsupdmmbllem 45238 smfinfdmmbllem 45242 |
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