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Mathbox for Glauco Siliprandi |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > salgenss | Structured version Visualization version GIF version |
Description: The sigma-algebra generated by a set is the smallest sigma-algebra, on the same base set, that includes the set. Proposition 111G (b) of [Fremlin1] p. 13. Notice that the condition "on the same base set" is needed, see the counterexample salgensscntex 41353, where a sigma-algebra is shown that includes a set, but does not include the sigma-algebra generated (the key is that its base set is larger than the base set of the generating set). (Contributed by Glauco Siliprandi, 3-Jan-2021.) |
Ref | Expression |
---|---|
salgenss.x | ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
salgenss.g | ⊢ 𝐺 = (SalGen‘𝑋) |
salgenss.s | ⊢ (𝜑 → 𝑆 ∈ SAlg) |
salgenss.i | ⊢ (𝜑 → 𝑋 ⊆ 𝑆) |
salgenss.u | ⊢ (𝜑 → ∪ 𝑆 = ∪ 𝑋) |
Ref | Expression |
---|---|
salgenss | ⊢ (𝜑 → 𝐺 ⊆ 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | salgenss.g | . . . 4 ⊢ 𝐺 = (SalGen‘𝑋) | |
2 | 1 | a1i 11 | . . 3 ⊢ (𝜑 → 𝐺 = (SalGen‘𝑋)) |
3 | salgenss.x | . . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
4 | salgenval 41332 | . . . 4 ⊢ (𝑋 ∈ 𝑉 → (SalGen‘𝑋) = ∩ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)}) | |
5 | 3, 4 | syl 17 | . . 3 ⊢ (𝜑 → (SalGen‘𝑋) = ∩ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)}) |
6 | 2, 5 | eqtrd 2861 | . 2 ⊢ (𝜑 → 𝐺 = ∩ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)}) |
7 | salgenss.s | . . . . 5 ⊢ (𝜑 → 𝑆 ∈ SAlg) | |
8 | salgenss.u | . . . . . 6 ⊢ (𝜑 → ∪ 𝑆 = ∪ 𝑋) | |
9 | salgenss.i | . . . . . 6 ⊢ (𝜑 → 𝑋 ⊆ 𝑆) | |
10 | 8, 9 | jca 509 | . . . . 5 ⊢ (𝜑 → (∪ 𝑆 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑆)) |
11 | 7, 10 | jca 509 | . . . 4 ⊢ (𝜑 → (𝑆 ∈ SAlg ∧ (∪ 𝑆 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑆))) |
12 | unieq 4666 | . . . . . . 7 ⊢ (𝑠 = 𝑆 → ∪ 𝑠 = ∪ 𝑆) | |
13 | 12 | eqeq1d 2827 | . . . . . 6 ⊢ (𝑠 = 𝑆 → (∪ 𝑠 = ∪ 𝑋 ↔ ∪ 𝑆 = ∪ 𝑋)) |
14 | sseq2 3852 | . . . . . 6 ⊢ (𝑠 = 𝑆 → (𝑋 ⊆ 𝑠 ↔ 𝑋 ⊆ 𝑆)) | |
15 | 13, 14 | anbi12d 626 | . . . . 5 ⊢ (𝑠 = 𝑆 → ((∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠) ↔ (∪ 𝑆 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑆))) |
16 | 15 | elrab 3585 | . . . 4 ⊢ (𝑆 ∈ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)} ↔ (𝑆 ∈ SAlg ∧ (∪ 𝑆 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑆))) |
17 | 11, 16 | sylibr 226 | . . 3 ⊢ (𝜑 → 𝑆 ∈ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)}) |
18 | intss1 4712 | . . 3 ⊢ (𝑆 ∈ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)} → ∩ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)} ⊆ 𝑆) | |
19 | 17, 18 | syl 17 | . 2 ⊢ (𝜑 → ∩ {𝑠 ∈ SAlg ∣ (∪ 𝑠 = ∪ 𝑋 ∧ 𝑋 ⊆ 𝑠)} ⊆ 𝑆) |
20 | 6, 19 | eqsstrd 3864 | 1 ⊢ (𝜑 → 𝐺 ⊆ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1658 ∈ wcel 2166 {crab 3121 ⊆ wss 3798 ∪ cuni 4658 ∩ cint 4697 ‘cfv 6123 SAlgcsalg 41319 SalGencsalgen 41323 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4659 df-int 4698 df-br 4874 df-opab 4936 df-mpt 4953 df-id 5250 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-iota 6086 df-fun 6125 df-fv 6131 df-salg 41320 df-salgen 41324 |
This theorem is referenced by: issalgend 41347 dfsalgen2 41350 borelmbl 41644 smfpimbor1lem2 41800 |
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