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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > rrvdmss | Structured version Visualization version GIF version |
Description: The domain of a random variable. This is useful to shorten proofs. (Contributed by Thierry Arnoux, 25-Jan-2017.) |
Ref | Expression |
---|---|
isrrvv.1 | ⊢ (𝜑 → 𝑃 ∈ Prob) |
rrvvf.1 | ⊢ (𝜑 → 𝑋 ∈ (rRndVar‘𝑃)) |
Ref | Expression |
---|---|
rrvdmss | ⊢ (𝜑 → ∪ dom 𝑃 ⊆ dom 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isrrvv.1 | . . 3 ⊢ (𝜑 → 𝑃 ∈ Prob) | |
2 | rrvvf.1 | . . 3 ⊢ (𝜑 → 𝑋 ∈ (rRndVar‘𝑃)) | |
3 | 1, 2 | rrvdm 34290 | . 2 ⊢ (𝜑 → dom 𝑋 = ∪ dom 𝑃) |
4 | eqimss2 4038 | . 2 ⊢ (dom 𝑋 = ∪ dom 𝑃 → ∪ dom 𝑃 ⊆ dom 𝑋) | |
5 | 3, 4 | syl 17 | 1 ⊢ (𝜑 → ∪ dom 𝑃 ⊆ dom 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1534 ∈ wcel 2099 ⊆ wss 3946 ∪ cuni 4905 dom cdm 5672 ‘cfv 6543 Probcprb 34251 rRndVarcrrv 34284 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5294 ax-nul 5301 ax-pow 5359 ax-pr 5423 ax-un 7735 ax-cnex 11202 ax-resscn 11203 ax-pre-lttri 11220 ax-pre-lttrn 11221 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-nel 3037 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4323 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-int 4947 df-iun 4995 df-br 5144 df-opab 5206 df-mpt 5227 df-id 5570 df-po 5584 df-so 5585 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7416 df-oprab 7417 df-mpo 7418 df-1st 7992 df-2nd 7993 df-er 8723 df-map 8846 df-en 8964 df-dom 8965 df-sdom 8966 df-pnf 11288 df-mnf 11289 df-xr 11290 df-ltxr 11291 df-le 11292 df-ioo 13373 df-topgen 17450 df-top 22881 df-bases 22934 df-esum 33871 df-siga 33952 df-sigagen 33982 df-brsiga 34025 df-meas 34039 df-mbfm 34093 df-prob 34252 df-rrv 34285 |
This theorem is referenced by: (None) |
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