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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 3256 | . . 3 ⊢ (𝜑 → ∀𝑘 ∈ 𝐾 𝐸 ∈ 𝑆) |
4 | dfiun3g 5981 | . . 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 2735 | . . . . 5 ⊢ (𝑘 ∈ 𝐾 ↦ 𝐸) = (𝑘 ∈ 𝐾 ↦ 𝐸) | |
10 | 1, 7, 8, 9, 2 | rnmptssdff 45221 | . . . 4 ⊢ (𝜑 → ran (𝑘 ∈ 𝐾 ↦ 𝐸) ⊆ 𝑆) |
11 | 6, 10 | sselpwd 5334 | . . 3 ⊢ (𝜑 → ran (𝑘 ∈ 𝐾 ↦ 𝐸) ∈ 𝒫 𝑆) |
12 | saliunclf.5 | . . . 4 ⊢ (𝜑 → 𝐾 ≼ ω) | |
13 | 7 | rn1st 45219 | . . . 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 46272 | . 2 ⊢ (𝜑 → ∪ ran (𝑘 ∈ 𝐾 ↦ 𝐸) ∈ 𝑆) |
16 | 5, 15 | eqeltrd 2839 | 1 ⊢ (𝜑 → ∪ 𝑘 ∈ 𝐾 𝐸 ∈ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 Ⅎwnf 1780 ∈ wcel 2106 Ⅎwnfc 2888 ∀wral 3059 ∪ cuni 4912 ∪ ciun 4996 class class class wbr 5148 ↦ cmpt 5231 ran crn 5690 ωcom 7887 ≼ cdom 8982 SAlgcsalg 46264 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-isom 6572 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-er 8744 df-map 8867 df-en 8985 df-dom 8986 df-card 9977 df-acn 9980 df-salg 46265 |
This theorem is referenced by: saliuncl 46279 saliinclf 46282 smfsupdmmbllem 46800 smfinfdmmbllem 46804 |
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