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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dmsigagen | Structured version Visualization version GIF version |
Description: A sigma-algebra can be generated from any set. (Contributed by Thierry Arnoux, 21-Jan-2017.) |
Ref | Expression |
---|---|
dmsigagen | ⊢ dom sigaGen = V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vuniex 7758 | . . . . . 6 ⊢ ∪ 𝑗 ∈ V | |
2 | pwsiga 34111 | . . . . . 6 ⊢ (∪ 𝑗 ∈ V → 𝒫 ∪ 𝑗 ∈ (sigAlgebra‘∪ 𝑗)) | |
3 | 1, 2 | ax-mp 5 | . . . . 5 ⊢ 𝒫 ∪ 𝑗 ∈ (sigAlgebra‘∪ 𝑗) |
4 | pwuni 4950 | . . . . 5 ⊢ 𝑗 ⊆ 𝒫 ∪ 𝑗 | |
5 | sseq2 4022 | . . . . . 6 ⊢ (𝑠 = 𝒫 ∪ 𝑗 → (𝑗 ⊆ 𝑠 ↔ 𝑗 ⊆ 𝒫 ∪ 𝑗)) | |
6 | 5 | rspcev 3622 | . . . . 5 ⊢ ((𝒫 ∪ 𝑗 ∈ (sigAlgebra‘∪ 𝑗) ∧ 𝑗 ⊆ 𝒫 ∪ 𝑗) → ∃𝑠 ∈ (sigAlgebra‘∪ 𝑗)𝑗 ⊆ 𝑠) |
7 | 3, 4, 6 | mp2an 692 | . . . 4 ⊢ ∃𝑠 ∈ (sigAlgebra‘∪ 𝑗)𝑗 ⊆ 𝑠 |
8 | rabn0 4395 | . . . 4 ⊢ ({𝑠 ∈ (sigAlgebra‘∪ 𝑗) ∣ 𝑗 ⊆ 𝑠} ≠ ∅ ↔ ∃𝑠 ∈ (sigAlgebra‘∪ 𝑗)𝑗 ⊆ 𝑠) | |
9 | 7, 8 | mpbir 231 | . . 3 ⊢ {𝑠 ∈ (sigAlgebra‘∪ 𝑗) ∣ 𝑗 ⊆ 𝑠} ≠ ∅ |
10 | intex 5350 | . . 3 ⊢ ({𝑠 ∈ (sigAlgebra‘∪ 𝑗) ∣ 𝑗 ⊆ 𝑠} ≠ ∅ ↔ ∩ {𝑠 ∈ (sigAlgebra‘∪ 𝑗) ∣ 𝑗 ⊆ 𝑠} ∈ V) | |
11 | 9, 10 | mpbi 230 | . 2 ⊢ ∩ {𝑠 ∈ (sigAlgebra‘∪ 𝑗) ∣ 𝑗 ⊆ 𝑠} ∈ V |
12 | df-sigagen 34120 | . 2 ⊢ sigaGen = (𝑗 ∈ V ↦ ∩ {𝑠 ∈ (sigAlgebra‘∪ 𝑗) ∣ 𝑗 ⊆ 𝑠}) | |
13 | 11, 12 | dmmpti 6713 | 1 ⊢ dom sigaGen = V |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2106 ≠ wne 2938 ∃wrex 3068 {crab 3433 Vcvv 3478 ⊆ wss 3963 ∅c0 4339 𝒫 cpw 4605 ∪ cuni 4912 ∩ cint 4951 dom cdm 5689 ‘cfv 6563 sigAlgebracsiga 34089 sigaGencsigagen 34119 |
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-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-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-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 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-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-iota 6516 df-fun 6565 df-fn 6566 df-fv 6571 df-siga 34090 df-sigagen 34120 |
This theorem is referenced by: (None) |
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