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Mirrors > Home > MPE Home > Th. List > Mathboxes > unisg | Structured version Visualization version GIF version |
Description: The sigma-algebra generated by a collection 𝐴 is a sigma-algebra on ∪ 𝐴. (Contributed by Thierry Arnoux, 27-Dec-2016.) |
Ref | Expression |
---|---|
unisg | ⊢ (𝐴 ∈ 𝑉 → ∪ (sigaGen‘𝐴) = ∪ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sigagensiga 33628 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → (sigaGen‘𝐴) ∈ (sigAlgebra‘∪ 𝐴)) | |
2 | issgon 33610 | . . . 4 ⊢ ((sigaGen‘𝐴) ∈ (sigAlgebra‘∪ 𝐴) ↔ ((sigaGen‘𝐴) ∈ ∪ ran sigAlgebra ∧ ∪ 𝐴 = ∪ (sigaGen‘𝐴))) | |
3 | 1, 2 | sylib 217 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ((sigaGen‘𝐴) ∈ ∪ ran sigAlgebra ∧ ∪ 𝐴 = ∪ (sigaGen‘𝐴))) |
4 | 3 | simprd 495 | . 2 ⊢ (𝐴 ∈ 𝑉 → ∪ 𝐴 = ∪ (sigaGen‘𝐴)) |
5 | 4 | eqcomd 2730 | 1 ⊢ (𝐴 ∈ 𝑉 → ∪ (sigaGen‘𝐴) = ∪ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 ∪ cuni 4899 ran crn 5667 ‘cfv 6533 sigAlgebracsiga 33595 sigaGencsigagen 33625 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-int 4941 df-br 5139 df-opab 5201 df-mpt 5222 df-id 5564 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-iota 6485 df-fun 6535 df-fn 6536 df-fv 6541 df-siga 33596 df-sigagen 33626 |
This theorem is referenced by: unibrsiga 33673 sxsigon 33679 imambfm 33750 cnmbfm 33751 sibf0 33822 sibff 33824 sibfof 33828 sitgclg 33830 orvcval4 33948 |
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