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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > saliuncl | Structured version Visualization version GIF version |
Description: SAlg sigma-algebra is closed under countable indexed union. (Contributed by Glauco Siliprandi, 17-Aug-2020.) |
Ref | Expression |
---|---|
saliuncl.s | ⊢ (𝜑 → 𝑆 ∈ SAlg) |
saliuncl.kct | ⊢ (𝜑 → 𝐾 ≼ ω) |
saliuncl.b | ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐾) → 𝐸 ∈ 𝑆) |
Ref | Expression |
---|---|
saliuncl | ⊢ (𝜑 → ∪ 𝑘 ∈ 𝐾 𝐸 ∈ 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | saliuncl.b | . . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ 𝐾) → 𝐸 ∈ 𝑆) | |
2 | 1 | ralrimiva 3148 | . . 3 ⊢ (𝜑 → ∀𝑘 ∈ 𝐾 𝐸 ∈ 𝑆) |
3 | dfiun3g 5624 | . . 3 ⊢ (∀𝑘 ∈ 𝐾 𝐸 ∈ 𝑆 → ∪ 𝑘 ∈ 𝐾 𝐸 = ∪ ran (𝑘 ∈ 𝐾 ↦ 𝐸)) | |
4 | 2, 3 | syl 17 | . 2 ⊢ (𝜑 → ∪ 𝑘 ∈ 𝐾 𝐸 = ∪ ran (𝑘 ∈ 𝐾 ↦ 𝐸)) |
5 | saliuncl.s | . . 3 ⊢ (𝜑 → 𝑆 ∈ SAlg) | |
6 | eqid 2778 | . . . . . 6 ⊢ (𝑘 ∈ 𝐾 ↦ 𝐸) = (𝑘 ∈ 𝐾 ↦ 𝐸) | |
7 | 6 | rnmptss 6656 | . . . . 5 ⊢ (∀𝑘 ∈ 𝐾 𝐸 ∈ 𝑆 → ran (𝑘 ∈ 𝐾 ↦ 𝐸) ⊆ 𝑆) |
8 | 2, 7 | syl 17 | . . . 4 ⊢ (𝜑 → ran (𝑘 ∈ 𝐾 ↦ 𝐸) ⊆ 𝑆) |
9 | 5, 8 | ssexd 5042 | . . . . 5 ⊢ (𝜑 → ran (𝑘 ∈ 𝐾 ↦ 𝐸) ∈ V) |
10 | elpwg 4387 | . . . . 5 ⊢ (ran (𝑘 ∈ 𝐾 ↦ 𝐸) ∈ V → (ran (𝑘 ∈ 𝐾 ↦ 𝐸) ∈ 𝒫 𝑆 ↔ ran (𝑘 ∈ 𝐾 ↦ 𝐸) ⊆ 𝑆)) | |
11 | 9, 10 | syl 17 | . . . 4 ⊢ (𝜑 → (ran (𝑘 ∈ 𝐾 ↦ 𝐸) ∈ 𝒫 𝑆 ↔ ran (𝑘 ∈ 𝐾 ↦ 𝐸) ⊆ 𝑆)) |
12 | 8, 11 | mpbird 249 | . . 3 ⊢ (𝜑 → ran (𝑘 ∈ 𝐾 ↦ 𝐸) ∈ 𝒫 𝑆) |
13 | saliuncl.kct | . . . 4 ⊢ (𝜑 → 𝐾 ≼ ω) | |
14 | 1stcrestlem 21664 | . . . 4 ⊢ (𝐾 ≼ ω → ran (𝑘 ∈ 𝐾 ↦ 𝐸) ≼ ω) | |
15 | 13, 14 | syl 17 | . . 3 ⊢ (𝜑 → ran (𝑘 ∈ 𝐾 ↦ 𝐸) ≼ ω) |
16 | 5, 12, 15 | salunicl 41460 | . 2 ⊢ (𝜑 → ∪ ran (𝑘 ∈ 𝐾 ↦ 𝐸) ∈ 𝑆) |
17 | 4, 16 | eqeltrd 2859 | 1 ⊢ (𝜑 → ∪ 𝑘 ∈ 𝐾 𝐸 ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 ∧ wa 386 = wceq 1601 ∈ wcel 2107 ∀wral 3090 Vcvv 3398 ⊆ wss 3792 𝒫 cpw 4379 ∪ cuni 4671 ∪ ciun 4753 class class class wbr 4886 ↦ cmpt 4965 ran crn 5356 ωcom 7343 ≼ cdom 8239 SAlgcsalg 41452 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-ral 3095 df-rex 3096 df-reu 3097 df-rmo 3098 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-pss 3808 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-tp 4403 df-op 4405 df-uni 4672 df-int 4711 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-se 5315 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-isom 6144 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-1st 7445 df-2nd 7446 df-wrecs 7689 df-recs 7751 df-er 8026 df-map 8142 df-en 8242 df-dom 8243 df-card 9098 df-acn 9101 df-salg 41453 |
This theorem is referenced by: saliincl 41469 subsaliuncl 41500 meaiunlelem 41609 meaiuninclem 41621 meaiuninc3v 41625 meaiininclem 41627 caratheodory 41669 opnvonmbllem2 41774 ctvonmbl 41830 vonct 41834 smfaddlem2 41899 smflimlem1 41906 smfresal 41922 smfmullem4 41928 |
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