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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > dstfrvel | Structured version Visualization version GIF version |
Description: Elementhood of preimage maps produced by the "less than or equal to" relation. (Contributed by Thierry Arnoux, 13-Feb-2017.) |
Ref | Expression |
---|---|
dstfrv.1 | ⊢ (𝜑 → 𝑃 ∈ Prob) |
dstfrv.2 | ⊢ (𝜑 → 𝑋 ∈ (rRndVar‘𝑃)) |
orvclteel.1 | ⊢ (𝜑 → 𝐴 ∈ ℝ) |
dstfrvel.1 | ⊢ (𝜑 → 𝐵 ∈ ∪ dom 𝑃) |
dstfrvel.2 | ⊢ (𝜑 → (𝑋‘𝐵) ≤ 𝐴) |
Ref | Expression |
---|---|
dstfrvel | ⊢ (𝜑 → 𝐵 ∈ (𝑋∘RV/𝑐 ≤ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dstfrv.1 | . . . . . 6 ⊢ (𝜑 → 𝑃 ∈ Prob) | |
2 | dstfrv.2 | . . . . . 6 ⊢ (𝜑 → 𝑋 ∈ (rRndVar‘𝑃)) | |
3 | 1, 2 | rrvvf 31812 | . . . . 5 ⊢ (𝜑 → 𝑋:∪ dom 𝑃⟶ℝ) |
4 | dstfrvel.1 | . . . . 5 ⊢ (𝜑 → 𝐵 ∈ ∪ dom 𝑃) | |
5 | 3, 4 | ffvelrnd 6829 | . . . 4 ⊢ (𝜑 → (𝑋‘𝐵) ∈ ℝ) |
6 | dstfrvel.2 | . . . 4 ⊢ (𝜑 → (𝑋‘𝐵) ≤ 𝐴) | |
7 | breq1 5033 | . . . . 5 ⊢ (𝑥 = (𝑋‘𝐵) → (𝑥 ≤ 𝐴 ↔ (𝑋‘𝐵) ≤ 𝐴)) | |
8 | 7 | elrab 3628 | . . . 4 ⊢ ((𝑋‘𝐵) ∈ {𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐴} ↔ ((𝑋‘𝐵) ∈ ℝ ∧ (𝑋‘𝐵) ≤ 𝐴)) |
9 | 5, 6, 8 | sylanbrc 586 | . . 3 ⊢ (𝜑 → (𝑋‘𝐵) ∈ {𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐴}) |
10 | 3 | ffund 6491 | . . . 4 ⊢ (𝜑 → Fun 𝑋) |
11 | 1, 2 | rrvdm 31814 | . . . . 5 ⊢ (𝜑 → dom 𝑋 = ∪ dom 𝑃) |
12 | 4, 11 | eleqtrrd 2893 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ dom 𝑋) |
13 | fvimacnv 6800 | . . . 4 ⊢ ((Fun 𝑋 ∧ 𝐵 ∈ dom 𝑋) → ((𝑋‘𝐵) ∈ {𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐴} ↔ 𝐵 ∈ (◡𝑋 “ {𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐴}))) | |
14 | 10, 12, 13 | syl2anc 587 | . . 3 ⊢ (𝜑 → ((𝑋‘𝐵) ∈ {𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐴} ↔ 𝐵 ∈ (◡𝑋 “ {𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐴}))) |
15 | 9, 14 | mpbid 235 | . 2 ⊢ (𝜑 → 𝐵 ∈ (◡𝑋 “ {𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐴})) |
16 | orvclteel.1 | . . 3 ⊢ (𝜑 → 𝐴 ∈ ℝ) | |
17 | 1, 2, 16 | orrvcval4 31832 | . 2 ⊢ (𝜑 → (𝑋∘RV/𝑐 ≤ 𝐴) = (◡𝑋 “ {𝑥 ∈ ℝ ∣ 𝑥 ≤ 𝐴})) |
18 | 15, 17 | eleqtrrd 2893 | 1 ⊢ (𝜑 → 𝐵 ∈ (𝑋∘RV/𝑐 ≤ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∈ wcel 2111 {crab 3110 ∪ cuni 4800 class class class wbr 5030 ◡ccnv 5518 dom cdm 5519 “ cima 5522 Fun wfun 6318 ‘cfv 6324 (class class class)co 7135 ℝcr 10525 ≤ cle 10665 Probcprb 31775 rRndVarcrrv 31808 ∘RV/𝑐corvc 31823 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-cnex 10582 ax-resscn 10583 ax-pre-lttri 10600 ax-pre-lttrn 10601 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-nel 3092 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-po 5438 df-so 5439 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-1st 7671 df-2nd 7672 df-er 8272 df-map 8391 df-en 8493 df-dom 8494 df-sdom 8495 df-pnf 10666 df-mnf 10667 df-xr 10668 df-ltxr 10669 df-le 10670 df-ioo 12730 df-topgen 16709 df-top 21499 df-bases 21551 df-esum 31397 df-siga 31478 df-sigagen 31508 df-brsiga 31551 df-meas 31565 df-mbfm 31619 df-prob 31776 df-rrv 31809 df-orvc 31824 |
This theorem is referenced by: dstfrvunirn 31842 |
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