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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 34282 | . . . . 5 ⊢ (𝜑 → dom 𝑋 = ∪ dom 𝑃) |
4 | 3 | eleq2d 2812 | . . . 4 ⊢ (𝜑 → (𝑧 ∈ dom 𝑋 ↔ 𝑧 ∈ ∪ dom 𝑃)) |
5 | 4 | biimprd 247 | . . 3 ⊢ (𝜑 → (𝑧 ∈ ∪ dom 𝑃 → 𝑧 ∈ dom 𝑋)) |
6 | 5 | imdistani 567 | . 2 ⊢ ((𝜑 ∧ 𝑧 ∈ ∪ dom 𝑃) → (𝜑 ∧ 𝑧 ∈ dom 𝑋)) |
7 | 1, 2 | rrvfn 34281 | . . . 4 ⊢ (𝜑 → 𝑋 Fn ∪ dom 𝑃) |
8 | fnfun 6662 | . . . 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 34295 | . 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 205 ∧ wa 394 ∈ wcel 2099 ∪ cuni 4915 class class class wbr 5155 dom cdm 5684 Fun wfun 6550 Fn wfn 6551 ‘cfv 6556 (class class class)co 7426 Probcprb 34243 rRndVarcrrv 34276 ∘RV/𝑐corvc 34291 |
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 5306 ax-nul 5313 ax-pow 5371 ax-pr 5435 ax-un 7748 ax-cnex 11216 ax-resscn 11217 ax-pre-lttri 11234 ax-pre-lttrn 11235 |
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 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4916 df-int 4957 df-iun 5005 df-br 5156 df-opab 5218 df-mpt 5239 df-id 5582 df-po 5596 df-so 5597 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6508 df-fun 6558 df-fn 6559 df-f 6560 df-f1 6561 df-fo 6562 df-f1o 6563 df-fv 6564 df-ov 7429 df-oprab 7430 df-mpo 7431 df-1st 8005 df-2nd 8006 df-er 8736 df-map 8859 df-en 8977 df-dom 8978 df-sdom 8979 df-pnf 11302 df-mnf 11303 df-xr 11304 df-ltxr 11305 df-le 11306 df-ioo 13384 df-topgen 17460 df-top 22890 df-bases 22943 df-esum 33863 df-siga 33944 df-sigagen 33974 df-brsiga 34017 df-meas 34031 df-mbfm 34085 df-prob 34244 df-rrv 34277 df-orvc 34292 |
This theorem is referenced by: (None) |
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