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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 31053 | . . . . 5 ⊢ (𝜑 → dom 𝑋 = ∪ dom 𝑃) |
4 | 3 | eleq2d 2891 | . . . 4 ⊢ (𝜑 → (𝑧 ∈ dom 𝑋 ↔ 𝑧 ∈ ∪ dom 𝑃)) |
5 | 4 | biimprd 240 | . . 3 ⊢ (𝜑 → (𝑧 ∈ ∪ dom 𝑃 → 𝑧 ∈ dom 𝑋)) |
6 | 5 | imdistani 566 | . 2 ⊢ ((𝜑 ∧ 𝑧 ∈ ∪ dom 𝑃) → (𝜑 ∧ 𝑧 ∈ dom 𝑋)) |
7 | 1, 2 | rrvfn 31052 | . . . 4 ⊢ (𝜑 → 𝑋 Fn ∪ dom 𝑃) |
8 | fnfun 6220 | . . . 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 31066 | . 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 198 ∧ wa 386 ∈ wcel 2166 ∪ cuni 4657 class class class wbr 4872 dom cdm 5341 Fun wfun 6116 Fn wfn 6117 ‘cfv 6122 (class class class)co 6904 Probcprb 31014 rRndVarcrrv 31047 ∘RV/𝑐corvc 31062 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2390 ax-ext 2802 ax-sep 5004 ax-nul 5012 ax-pow 5064 ax-pr 5126 ax-un 7208 ax-cnex 10307 ax-resscn 10308 ax-pre-lttri 10325 ax-pre-lttrn 10326 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-fal 1672 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2604 df-eu 2639 df-clab 2811 df-cleq 2817 df-clel 2820 df-nfc 2957 df-ne 2999 df-nel 3102 df-ral 3121 df-rex 3122 df-rab 3125 df-v 3415 df-sbc 3662 df-csb 3757 df-dif 3800 df-un 3802 df-in 3804 df-ss 3811 df-nul 4144 df-if 4306 df-pw 4379 df-sn 4397 df-pr 4399 df-op 4403 df-uni 4658 df-int 4697 df-iun 4741 df-br 4873 df-opab 4935 df-mpt 4952 df-id 5249 df-po 5262 df-so 5263 df-xp 5347 df-rel 5348 df-cnv 5349 df-co 5350 df-dm 5351 df-rn 5352 df-res 5353 df-ima 5354 df-iota 6085 df-fun 6124 df-fn 6125 df-f 6126 df-f1 6127 df-fo 6128 df-f1o 6129 df-fv 6130 df-ov 6907 df-oprab 6908 df-mpt2 6909 df-1st 7427 df-2nd 7428 df-er 8008 df-map 8123 df-en 8222 df-dom 8223 df-sdom 8224 df-pnf 10392 df-mnf 10393 df-xr 10394 df-ltxr 10395 df-le 10396 df-ioo 12466 df-topgen 16456 df-top 21068 df-bases 21120 df-esum 30634 df-siga 30715 df-sigagen 30746 df-brsiga 30789 df-meas 30803 df-mbfm 30857 df-prob 31015 df-rrv 31048 df-orvc 31063 |
This theorem is referenced by: (None) |
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