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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > elorrvc | Structured version Visualization version GIF version |
Description: Elementhood of a preimage for a real-valued random variable. (Contributed by Thierry Arnoux, 21-Jan-2017.) |
Ref | Expression |
---|---|
orrvccel.1 | ⊢ (𝜑 → 𝑃 ∈ Prob) |
orrvccel.2 | ⊢ (𝜑 → 𝑋 ∈ (rRndVar‘𝑃)) |
orrvccel.4 | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
Ref | Expression |
---|---|
elorrvc | ⊢ ((𝜑 ∧ 𝑧 ∈ ∪ dom 𝑃) → (𝑧 ∈ (𝑋∘RV/𝑐𝑅𝐴) ↔ (𝑋‘𝑧)𝑅𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | orrvccel.1 | . . . . . 6 ⊢ (𝜑 → 𝑃 ∈ Prob) | |
2 | orrvccel.2 | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ (rRndVar‘𝑃)) | |
3 | 1, 2 | rrvdm 31814 | . . . . 5 ⊢ (𝜑 → dom 𝑋 = ∪ dom 𝑃) |
4 | 3 | eleq2d 2875 | . . . 4 ⊢ (𝜑 → (𝑧 ∈ dom 𝑋 ↔ 𝑧 ∈ ∪ dom 𝑃)) |
5 | 4 | biimprd 251 | . . 3 ⊢ (𝜑 → (𝑧 ∈ ∪ dom 𝑃 → 𝑧 ∈ dom 𝑋)) |
6 | 5 | imdistani 572 | . 2 ⊢ ((𝜑 ∧ 𝑧 ∈ ∪ dom 𝑃) → (𝜑 ∧ 𝑧 ∈ dom 𝑋)) |
7 | 1, 2 | rrvfn 31813 | . . . 4 ⊢ (𝜑 → 𝑋 Fn ∪ dom 𝑃) |
8 | fnfun 6423 | . . . 4 ⊢ (𝑋 Fn ∪ dom 𝑃 → Fun 𝑋) | |
9 | 7, 8 | syl 17 | . . 3 ⊢ (𝜑 → Fun 𝑋) |
10 | orrvccel.4 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
11 | 9, 2, 10 | elorvc 31827 | . 2 ⊢ ((𝜑 ∧ 𝑧 ∈ dom 𝑋) → (𝑧 ∈ (𝑋∘RV/𝑐𝑅𝐴) ↔ (𝑋‘𝑧)𝑅𝐴)) |
12 | 6, 11 | syl 17 | 1 ⊢ ((𝜑 ∧ 𝑧 ∈ ∪ dom 𝑃) → (𝑧 ∈ (𝑋∘RV/𝑐𝑅𝐴) ↔ (𝑋‘𝑧)𝑅𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∈ wcel 2111 ∪ cuni 4800 class class class wbr 5030 dom cdm 5519 Fun wfun 6318 Fn wfn 6319 ‘cfv 6324 (class class class)co 7135 Probcprb 31775 rRndVarcrrv 31808 ∘RV/𝑐corvc 31823 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-pre-lttri 10600 ax-pre-lttrn 10601 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-po 5438 df-so 5439 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-1st 7671 df-2nd 7672 df-er 8272 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-ioo 12730 df-topgen 16709 df-top 21499 df-bases 21551 df-esum 31397 df-siga 31478 df-sigagen 31508 df-brsiga 31551 df-meas 31565 df-mbfm 31619 df-prob 31776 df-rrv 31809 df-orvc 31824 |
This theorem is referenced by: (None) |
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