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| Description: The property of being a probability measure. (Contributed by Thierry Arnoux, 8-Dec-2016.) | 
| Ref | Expression | 
|---|---|
| elprob | ⊢ (𝑃 ∈ Prob ↔ (𝑃 ∈ ∪ ran measures ∧ (𝑃‘∪ dom 𝑃) = 1)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | id 22 | . . . 4 ⊢ (𝑝 = 𝑃 → 𝑝 = 𝑃) | |
| 2 | dmeq 5913 | . . . . 5 ⊢ (𝑝 = 𝑃 → dom 𝑝 = dom 𝑃) | |
| 3 | 2 | unieqd 4919 | . . . 4 ⊢ (𝑝 = 𝑃 → ∪ dom 𝑝 = ∪ dom 𝑃) | 
| 4 | 1, 3 | fveq12d 6912 | . . 3 ⊢ (𝑝 = 𝑃 → (𝑝‘∪ dom 𝑝) = (𝑃‘∪ dom 𝑃)) | 
| 5 | 4 | eqeq1d 2738 | . 2 ⊢ (𝑝 = 𝑃 → ((𝑝‘∪ dom 𝑝) = 1 ↔ (𝑃‘∪ dom 𝑃) = 1)) | 
| 6 | df-prob 34411 | . 2 ⊢ Prob = {𝑝 ∈ ∪ ran measures ∣ (𝑝‘∪ dom 𝑝) = 1} | |
| 7 | 5, 6 | elrab2 3694 | 1 ⊢ (𝑃 ∈ Prob ↔ (𝑃 ∈ ∪ ran measures ∧ (𝑃‘∪ dom 𝑃) = 1)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1539 ∈ wcel 2107 ∪ cuni 4906 dom cdm 5684 ran crn 5685 ‘cfv 6560 1c1 11157 measurescmeas 34197 Probcprb 34410 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-dm 5694 df-iota 6513 df-fv 6568 df-prob 34411 | 
| This theorem is referenced by: domprobmeas 34413 probtot 34415 probfinmeasb 34431 probfinmeasbALTV 34432 probmeasb 34433 dstrvprob 34475 | 
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