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 6852 | . . . . . 6 ⊢ ∪ 𝑗 ∈ V | |
2 | pwsiga 29520 | . . . . . 6 ⊢ (∪ 𝑗 ∈ V → 𝒫 ∪ 𝑗 ∈ (sigAlgebra‘∪ 𝑗)) | |
3 | 1, 2 | ax-mp 5 | . . . . 5 ⊢ 𝒫 ∪ 𝑗 ∈ (sigAlgebra‘∪ 𝑗) |
4 | pwuni 4825 | . . . . 5 ⊢ 𝑗 ⊆ 𝒫 ∪ 𝑗 | |
5 | sseq2 3590 | . . . . . 6 ⊢ (𝑠 = 𝒫 ∪ 𝑗 → (𝑗 ⊆ 𝑠 ↔ 𝑗 ⊆ 𝒫 ∪ 𝑗)) | |
6 | 5 | rspcev 3282 | . . . . 5 ⊢ ((𝒫 ∪ 𝑗 ∈ (sigAlgebra‘∪ 𝑗) ∧ 𝑗 ⊆ 𝒫 ∪ 𝑗) → ∃𝑠 ∈ (sigAlgebra‘∪ 𝑗)𝑗 ⊆ 𝑠) |
7 | 3, 4, 6 | mp2an 704 | . . . 4 ⊢ ∃𝑠 ∈ (sigAlgebra‘∪ 𝑗)𝑗 ⊆ 𝑠 |
8 | rabn0 3912 | . . . 4 ⊢ ({𝑠 ∈ (sigAlgebra‘∪ 𝑗) ∣ 𝑗 ⊆ 𝑠} ≠ ∅ ↔ ∃𝑠 ∈ (sigAlgebra‘∪ 𝑗)𝑗 ⊆ 𝑠) | |
9 | 7, 8 | mpbir 220 | . . 3 ⊢ {𝑠 ∈ (sigAlgebra‘∪ 𝑗) ∣ 𝑗 ⊆ 𝑠} ≠ ∅ |
10 | intex 4747 | . . 3 ⊢ ({𝑠 ∈ (sigAlgebra‘∪ 𝑗) ∣ 𝑗 ⊆ 𝑠} ≠ ∅ ↔ ∩ {𝑠 ∈ (sigAlgebra‘∪ 𝑗) ∣ 𝑗 ⊆ 𝑠} ∈ V) | |
11 | 9, 10 | mpbi 219 | . 2 ⊢ ∩ {𝑠 ∈ (sigAlgebra‘∪ 𝑗) ∣ 𝑗 ⊆ 𝑠} ∈ V |
12 | df-sigagen 29529 | . 2 ⊢ sigaGen = (𝑗 ∈ V ↦ ∩ {𝑠 ∈ (sigAlgebra‘∪ 𝑗) ∣ 𝑗 ⊆ 𝑠}) | |
13 | 11, 12 | dmmpti 5936 | 1 ⊢ dom sigaGen = V |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∃wrex 2897 {crab 2900 Vcvv 3173 ⊆ wss 3540 ∅c0 3874 𝒫 cpw 4108 ∪ cuni 4372 ∩ cint 4410 dom cdm 5038 ‘cfv 5804 sigAlgebracsiga 29497 sigaGencsigagen 29528 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-int 4411 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-iota 5768 df-fun 5806 df-fn 5807 df-fv 5812 df-siga 29498 df-sigagen 29529 |
This theorem is referenced by: (None) |
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