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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 3258 | . . 3 ⊢ (𝜑 → ∀𝑘 ∈ 𝐾 𝐸 ∈ 𝑆) | 
| 4 | dfiun3g 5978 | . . 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 2737 | . . . . 5 ⊢ (𝑘 ∈ 𝐾 ↦ 𝐸) = (𝑘 ∈ 𝐾 ↦ 𝐸) | |
| 10 | 1, 7, 8, 9, 2 | rnmptssdff 45282 | . . . 4 ⊢ (𝜑 → ran (𝑘 ∈ 𝐾 ↦ 𝐸) ⊆ 𝑆) | 
| 11 | 6, 10 | sselpwd 5328 | . . 3 ⊢ (𝜑 → ran (𝑘 ∈ 𝐾 ↦ 𝐸) ∈ 𝒫 𝑆) | 
| 12 | saliunclf.5 | . . . 4 ⊢ (𝜑 → 𝐾 ≼ ω) | |
| 13 | 7 | rn1st 45280 | . . . 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 46331 | . 2 ⊢ (𝜑 → ∪ ran (𝑘 ∈ 𝐾 ↦ 𝐸) ∈ 𝑆) | 
| 16 | 5, 15 | eqeltrd 2841 | 1 ⊢ (𝜑 → ∪ 𝑘 ∈ 𝐾 𝐸 ∈ 𝑆) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 Ⅎwnf 1783 ∈ wcel 2108 Ⅎwnfc 2890 ∀wral 3061 ∪ cuni 4907 ∪ ciun 4991 class class class wbr 5143 ↦ cmpt 5225 ran crn 5686 ωcom 7887 ≼ cdom 8983 SAlgcsalg 46323 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-se 5638 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-isom 6570 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8014 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-card 9979 df-acn 9982 df-salg 46324 | 
| This theorem is referenced by: saliuncl 46338 saliinclf 46341 smfsupdmmbllem 46859 smfinfdmmbllem 46863 | 
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