![]() |
Mathbox for Thierry Arnoux |
< Previous
Next >
Nearby theorems |
|
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 7712 | . . . . . 6 ⊢ ∪ 𝑗 ∈ V | |
2 | pwsiga 32959 | . . . . . 6 ⊢ (∪ 𝑗 ∈ V → 𝒫 ∪ 𝑗 ∈ (sigAlgebra‘∪ 𝑗)) | |
3 | 1, 2 | ax-mp 5 | . . . . 5 ⊢ 𝒫 ∪ 𝑗 ∈ (sigAlgebra‘∪ 𝑗) |
4 | pwuni 4942 | . . . . 5 ⊢ 𝑗 ⊆ 𝒫 ∪ 𝑗 | |
5 | sseq2 4004 | . . . . . 6 ⊢ (𝑠 = 𝒫 ∪ 𝑗 → (𝑗 ⊆ 𝑠 ↔ 𝑗 ⊆ 𝒫 ∪ 𝑗)) | |
6 | 5 | rspcev 3609 | . . . . 5 ⊢ ((𝒫 ∪ 𝑗 ∈ (sigAlgebra‘∪ 𝑗) ∧ 𝑗 ⊆ 𝒫 ∪ 𝑗) → ∃𝑠 ∈ (sigAlgebra‘∪ 𝑗)𝑗 ⊆ 𝑠) |
7 | 3, 4, 6 | mp2an 690 | . . . 4 ⊢ ∃𝑠 ∈ (sigAlgebra‘∪ 𝑗)𝑗 ⊆ 𝑠 |
8 | rabn0 4381 | . . . 4 ⊢ ({𝑠 ∈ (sigAlgebra‘∪ 𝑗) ∣ 𝑗 ⊆ 𝑠} ≠ ∅ ↔ ∃𝑠 ∈ (sigAlgebra‘∪ 𝑗)𝑗 ⊆ 𝑠) | |
9 | 7, 8 | mpbir 230 | . . 3 ⊢ {𝑠 ∈ (sigAlgebra‘∪ 𝑗) ∣ 𝑗 ⊆ 𝑠} ≠ ∅ |
10 | intex 5330 | . . 3 ⊢ ({𝑠 ∈ (sigAlgebra‘∪ 𝑗) ∣ 𝑗 ⊆ 𝑠} ≠ ∅ ↔ ∩ {𝑠 ∈ (sigAlgebra‘∪ 𝑗) ∣ 𝑗 ⊆ 𝑠} ∈ V) | |
11 | 9, 10 | mpbi 229 | . 2 ⊢ ∩ {𝑠 ∈ (sigAlgebra‘∪ 𝑗) ∣ 𝑗 ⊆ 𝑠} ∈ V |
12 | df-sigagen 32968 | . 2 ⊢ sigaGen = (𝑗 ∈ V ↦ ∩ {𝑠 ∈ (sigAlgebra‘∪ 𝑗) ∣ 𝑗 ⊆ 𝑠}) | |
13 | 11, 12 | dmmpti 6681 | 1 ⊢ dom sigaGen = V |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1541 ∈ wcel 2106 ≠ wne 2939 ∃wrex 3069 {crab 3431 Vcvv 3473 ⊆ wss 3944 ∅c0 4318 𝒫 cpw 4596 ∪ cuni 4901 ∩ cint 4943 dom cdm 5669 ‘cfv 6532 sigAlgebracsiga 32937 sigaGencsigagen 32967 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2702 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7708 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-int 4944 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-iota 6484 df-fun 6534 df-fn 6535 df-fv 6540 df-siga 32938 df-sigagen 32968 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |