Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > orrvcval4 | Structured version Visualization version GIF version |
Description: The value of the preimage mapping operator can be restricted to preimages of subsets of RR. (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 |
---|---|
orrvcval4 | ⊢ (𝜑 → (𝑋∘RV/𝑐𝑅𝐴) = (◡𝑋 “ {𝑦 ∈ ℝ ∣ 𝑦𝑅𝐴})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | orrvccel.1 | . . . 4 ⊢ (𝜑 → 𝑃 ∈ Prob) | |
2 | domprobsiga 31671 | . . . 4 ⊢ (𝑃 ∈ Prob → dom 𝑃 ∈ ∪ ran sigAlgebra) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (𝜑 → dom 𝑃 ∈ ∪ ran sigAlgebra) |
4 | retop 23372 | . . . 4 ⊢ (topGen‘ran (,)) ∈ Top | |
5 | 4 | a1i 11 | . . 3 ⊢ (𝜑 → (topGen‘ran (,)) ∈ Top) |
6 | orrvccel.2 | . . . . 5 ⊢ (𝜑 → 𝑋 ∈ (rRndVar‘𝑃)) | |
7 | 1 | rrvmbfm 31702 | . . . . 5 ⊢ (𝜑 → (𝑋 ∈ (rRndVar‘𝑃) ↔ 𝑋 ∈ (dom 𝑃MblFnM𝔅ℝ))) |
8 | 6, 7 | mpbid 234 | . . . 4 ⊢ (𝜑 → 𝑋 ∈ (dom 𝑃MblFnM𝔅ℝ)) |
9 | df-brsiga 31443 | . . . . 5 ⊢ 𝔅ℝ = (sigaGen‘(topGen‘ran (,))) | |
10 | 9 | oveq2i 7169 | . . . 4 ⊢ (dom 𝑃MblFnM𝔅ℝ) = (dom 𝑃MblFnM(sigaGen‘(topGen‘ran (,)))) |
11 | 8, 10 | eleqtrdi 2925 | . . 3 ⊢ (𝜑 → 𝑋 ∈ (dom 𝑃MblFnM(sigaGen‘(topGen‘ran (,))))) |
12 | orrvccel.4 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
13 | 3, 5, 11, 12 | orvcval4 31720 | . 2 ⊢ (𝜑 → (𝑋∘RV/𝑐𝑅𝐴) = (◡𝑋 “ {𝑦 ∈ ∪ (topGen‘ran (,)) ∣ 𝑦𝑅𝐴})) |
14 | uniretop 23373 | . . . 4 ⊢ ℝ = ∪ (topGen‘ran (,)) | |
15 | rabeq 3485 | . . . 4 ⊢ (ℝ = ∪ (topGen‘ran (,)) → {𝑦 ∈ ℝ ∣ 𝑦𝑅𝐴} = {𝑦 ∈ ∪ (topGen‘ran (,)) ∣ 𝑦𝑅𝐴}) | |
16 | 14, 15 | ax-mp 5 | . . 3 ⊢ {𝑦 ∈ ℝ ∣ 𝑦𝑅𝐴} = {𝑦 ∈ ∪ (topGen‘ran (,)) ∣ 𝑦𝑅𝐴} |
17 | 16 | imaeq2i 5929 | . 2 ⊢ (◡𝑋 “ {𝑦 ∈ ℝ ∣ 𝑦𝑅𝐴}) = (◡𝑋 “ {𝑦 ∈ ∪ (topGen‘ran (,)) ∣ 𝑦𝑅𝐴}) |
18 | 13, 17 | syl6eqr 2876 | 1 ⊢ (𝜑 → (𝑋∘RV/𝑐𝑅𝐴) = (◡𝑋 “ {𝑦 ∈ ℝ ∣ 𝑦𝑅𝐴})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 {crab 3144 ∪ cuni 4840 class class class wbr 5068 ◡ccnv 5556 dom cdm 5557 ran crn 5558 “ cima 5560 ‘cfv 6357 (class class class)co 7158 ℝcr 10538 (,)cioo 12741 topGenctg 16713 Topctop 21503 sigAlgebracsiga 31369 sigaGencsigagen 31399 𝔅ℝcbrsiga 31442 MblFnMcmbfm 31510 Probcprb 31667 rRndVarcrrv 31700 ∘RV/𝑐corvc 31715 |
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 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 ax-cnex 10595 ax-resscn 10596 ax-pre-lttri 10613 ax-pre-lttrn 10614 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-nel 3126 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-po 5476 df-so 5477 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-1st 7691 df-2nd 7692 df-er 8291 df-map 8410 df-en 8512 df-dom 8513 df-sdom 8514 df-pnf 10679 df-mnf 10680 df-xr 10681 df-ltxr 10682 df-le 10683 df-ioo 12745 df-topgen 16719 df-top 21504 df-bases 21556 df-esum 31289 df-siga 31370 df-sigagen 31400 df-brsiga 31443 df-meas 31457 df-mbfm 31511 df-prob 31668 df-rrv 31701 df-orvc 31716 |
This theorem is referenced by: orvcelval 31728 dstfrvel 31733 orvclteinc 31735 |
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