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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > unveldomd | Structured version Visualization version GIF version |
Description: The universe is an element of the domain of the probability, the universe (entire probability space) being ∪ dom 𝑃 in our construction. (Contributed by Thierry Arnoux, 22-Jan-2017.) |
Ref | Expression |
---|---|
unveldomd.1 | ⊢ (𝜑 → 𝑃 ∈ Prob) |
Ref | Expression |
---|---|
unveldomd | ⊢ (𝜑 → ∪ dom 𝑃 ∈ dom 𝑃) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | unveldomd.1 | . 2 ⊢ (𝜑 → 𝑃 ∈ Prob) | |
2 | domprobsiga 31072 | . 2 ⊢ (𝑃 ∈ Prob → dom 𝑃 ∈ ∪ ran sigAlgebra) | |
3 | sgon 30785 | . 2 ⊢ (dom 𝑃 ∈ ∪ ran sigAlgebra → dom 𝑃 ∈ (sigAlgebra‘∪ dom 𝑃)) | |
4 | baselsiga 30776 | . 2 ⊢ (dom 𝑃 ∈ (sigAlgebra‘∪ dom 𝑃) → ∪ dom 𝑃 ∈ dom 𝑃) | |
5 | 1, 2, 3, 4 | 4syl 19 | 1 ⊢ (𝜑 → ∪ dom 𝑃 ∈ dom 𝑃) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2107 ∪ cuni 4671 dom cdm 5355 ran crn 5356 ‘cfv 6135 sigAlgebracsiga 30768 Probcprb 31068 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-fal 1615 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4672 df-br 4887 df-opab 4949 df-mpt 4966 df-id 5261 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-fv 6143 df-ov 6925 df-esum 30688 df-siga 30769 df-meas 30857 df-prob 31069 |
This theorem is referenced by: unveldom 31077 probdsb 31083 probtotrnd 31086 cndprobtot 31097 0rrv 31112 rrvadd 31113 dstfrvclim1 31138 |
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