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| Mirrors > Home > MPE Home > Th. List > Mathboxes > orvcelel | Structured version Visualization version GIF version | ||
| Description: Preimage maps produced by the membership relation are measurable sets. (Contributed by Thierry Arnoux, 5-Feb-2017.) |
| Ref | Expression |
|---|---|
| dstrvprob.1 | ⊢ (𝜑 → 𝑃 ∈ Prob) |
| dstrvprob.2 | ⊢ (𝜑 → 𝑋 ∈ (rRndVar‘𝑃)) |
| orvcelel.1 | ⊢ (𝜑 → 𝐴 ∈ 𝔅ℝ) |
| Ref | Expression |
|---|---|
| orvcelel | ⊢ (𝜑 → (𝑋∘RV/𝑐 E 𝐴) ∈ dom 𝑃) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | dstrvprob.1 | . . 3 ⊢ (𝜑 → 𝑃 ∈ Prob) | |
| 2 | dstrvprob.2 | . . 3 ⊢ (𝜑 → 𝑋 ∈ (rRndVar‘𝑃)) | |
| 3 | orvcelel.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝔅ℝ) | |
| 4 | 1, 2, 3 | orvcelval 34471 | . 2 ⊢ (𝜑 → (𝑋∘RV/𝑐 E 𝐴) = (◡𝑋 “ 𝐴)) |
| 5 | 1, 2 | rrvfinvima 34452 | . . 3 ⊢ (𝜑 → ∀𝑎 ∈ 𝔅ℝ (◡𝑋 “ 𝑎) ∈ dom 𝑃) |
| 6 | simpr 484 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑎 = 𝐴) → 𝑎 = 𝐴) | |
| 7 | 6 | imaeq2d 6078 | . . . . 5 ⊢ ((𝜑 ∧ 𝑎 = 𝐴) → (◡𝑋 “ 𝑎) = (◡𝑋 “ 𝐴)) |
| 8 | 7 | eleq1d 2826 | . . . 4 ⊢ ((𝜑 ∧ 𝑎 = 𝐴) → ((◡𝑋 “ 𝑎) ∈ dom 𝑃 ↔ (◡𝑋 “ 𝐴) ∈ dom 𝑃)) |
| 9 | 3, 8 | rspcdv 3614 | . . 3 ⊢ (𝜑 → (∀𝑎 ∈ 𝔅ℝ (◡𝑋 “ 𝑎) ∈ dom 𝑃 → (◡𝑋 “ 𝐴) ∈ dom 𝑃)) |
| 10 | 5, 9 | mpd 15 | . 2 ⊢ (𝜑 → (◡𝑋 “ 𝐴) ∈ dom 𝑃) |
| 11 | 4, 10 | eqeltrd 2841 | 1 ⊢ (𝜑 → (𝑋∘RV/𝑐 E 𝐴) ∈ dom 𝑃) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∈ wcel 2108 ∀wral 3061 E cep 5583 ◡ccnv 5684 dom cdm 5685 “ cima 5688 ‘cfv 6561 (class class class)co 7431 𝔅ℝcbrsiga 34182 Probcprb 34409 rRndVarcrrv 34442 ∘RV/𝑐corvc 34458 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-cnex 11211 ax-resscn 11212 ax-pre-lttri 11229 ax-pre-lttrn 11230 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-nel 3047 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-int 4947 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-1st 8014 df-2nd 8015 df-er 8745 df-map 8868 df-en 8986 df-dom 8987 df-sdom 8988 df-pnf 11297 df-mnf 11298 df-xr 11299 df-ltxr 11300 df-le 11301 df-ioo 13391 df-topgen 17488 df-top 22900 df-bases 22953 df-esum 34029 df-siga 34110 df-sigagen 34140 df-brsiga 34183 df-meas 34197 df-mbfm 34251 df-prob 34410 df-rrv 34443 df-orvc 34459 |
| This theorem is referenced by: dstrvprob 34474 |
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