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 7465 | . . . . . 6 ⊢ ∪ 𝑗 ∈ V | |
2 | pwsiga 31389 | . . . . . 6 ⊢ (∪ 𝑗 ∈ V → 𝒫 ∪ 𝑗 ∈ (sigAlgebra‘∪ 𝑗)) | |
3 | 1, 2 | ax-mp 5 | . . . . 5 ⊢ 𝒫 ∪ 𝑗 ∈ (sigAlgebra‘∪ 𝑗) |
4 | pwuni 4875 | . . . . 5 ⊢ 𝑗 ⊆ 𝒫 ∪ 𝑗 | |
5 | sseq2 3993 | . . . . . 6 ⊢ (𝑠 = 𝒫 ∪ 𝑗 → (𝑗 ⊆ 𝑠 ↔ 𝑗 ⊆ 𝒫 ∪ 𝑗)) | |
6 | 5 | rspcev 3623 | . . . . 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 4339 | . . . 4 ⊢ ({𝑠 ∈ (sigAlgebra‘∪ 𝑗) ∣ 𝑗 ⊆ 𝑠} ≠ ∅ ↔ ∃𝑠 ∈ (sigAlgebra‘∪ 𝑗)𝑗 ⊆ 𝑠) | |
9 | 7, 8 | mpbir 233 | . . 3 ⊢ {𝑠 ∈ (sigAlgebra‘∪ 𝑗) ∣ 𝑗 ⊆ 𝑠} ≠ ∅ |
10 | intex 5240 | . . 3 ⊢ ({𝑠 ∈ (sigAlgebra‘∪ 𝑗) ∣ 𝑗 ⊆ 𝑠} ≠ ∅ ↔ ∩ {𝑠 ∈ (sigAlgebra‘∪ 𝑗) ∣ 𝑗 ⊆ 𝑠} ∈ V) | |
11 | 9, 10 | mpbi 232 | . 2 ⊢ ∩ {𝑠 ∈ (sigAlgebra‘∪ 𝑗) ∣ 𝑗 ⊆ 𝑠} ∈ V |
12 | df-sigagen 31398 | . 2 ⊢ sigaGen = (𝑗 ∈ V ↦ ∩ {𝑠 ∈ (sigAlgebra‘∪ 𝑗) ∣ 𝑗 ⊆ 𝑠}) | |
13 | 11, 12 | dmmpti 6492 | 1 ⊢ dom sigaGen = V |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∈ wcel 2114 ≠ wne 3016 ∃wrex 3139 {crab 3142 Vcvv 3494 ⊆ wss 3936 ∅c0 4291 𝒫 cpw 4539 ∪ cuni 4838 ∩ cint 4876 dom cdm 5555 ‘cfv 6355 sigAlgebracsiga 31367 sigaGencsigagen 31397 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-int 4877 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-iota 6314 df-fun 6357 df-fn 6358 df-fv 6363 df-siga 31368 df-sigagen 31398 |
This theorem is referenced by: (None) |
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