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Mirrors > Home > MPE Home > Th. List > Mathboxes > rrvvf | Structured version Visualization version GIF version |
Description: A real-valued random variable is a function. (Contributed by Thierry Arnoux, 25-Jan-2017.) |
Ref | Expression |
---|---|
isrrvv.1 | ⊢ (𝜑 → 𝑃 ∈ Prob) |
rrvvf.1 | ⊢ (𝜑 → 𝑋 ∈ (rRndVar‘𝑃)) |
Ref | Expression |
---|---|
rrvvf | ⊢ (𝜑 → 𝑋:∪ dom 𝑃⟶ℝ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | rrvvf.1 | . . 3 ⊢ (𝜑 → 𝑋 ∈ (rRndVar‘𝑃)) | |
2 | isrrvv.1 | . . . 4 ⊢ (𝜑 → 𝑃 ∈ Prob) | |
3 | 2 | isrrvv 31701 | . . 3 ⊢ (𝜑 → (𝑋 ∈ (rRndVar‘𝑃) ↔ (𝑋:∪ dom 𝑃⟶ℝ ∧ ∀𝑦 ∈ 𝔅ℝ (◡𝑋 “ 𝑦) ∈ dom 𝑃))) |
4 | 1, 3 | mpbid 234 | . 2 ⊢ (𝜑 → (𝑋:∪ dom 𝑃⟶ℝ ∧ ∀𝑦 ∈ 𝔅ℝ (◡𝑋 “ 𝑦) ∈ dom 𝑃)) |
5 | 4 | simpld 497 | 1 ⊢ (𝜑 → 𝑋:∪ dom 𝑃⟶ℝ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∈ wcel 2114 ∀wral 3138 ∪ cuni 4838 ◡ccnv 5554 dom cdm 5555 “ cima 5558 ⟶wf 6351 ‘cfv 6355 ℝcr 10536 𝔅ℝcbrsiga 31440 Probcprb 31665 rRndVarcrrv 31698 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-cnex 10593 ax-resscn 10594 ax-pre-lttri 10611 ax-pre-lttrn 10612 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-nel 3124 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-po 5474 df-so 5475 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-ov 7159 df-oprab 7160 df-mpo 7161 df-1st 7689 df-2nd 7690 df-er 8289 df-map 8408 df-en 8510 df-dom 8511 df-sdom 8512 df-pnf 10677 df-mnf 10678 df-xr 10679 df-ltxr 10680 df-le 10681 df-ioo 12743 df-topgen 16717 df-top 21502 df-bases 21554 df-esum 31287 df-siga 31368 df-sigagen 31398 df-brsiga 31441 df-meas 31455 df-mbfm 31509 df-prob 31666 df-rrv 31699 |
This theorem is referenced by: rrvfn 31703 rrvdm 31704 rrvrnss 31705 rrvf2 31706 rrvadd 31710 rrvmulc 31711 dstrvprob 31729 dstfrvel 31731 dstfrvunirn 31732 |
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