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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > orvcelval | Structured version Visualization version GIF version |
Description: Preimage maps produced by the membership relation. (Contributed by Thierry Arnoux, 6-Feb-2017.) |
Ref | Expression |
---|---|
dstrvprob.1 | ⊢ (𝜑 → 𝑃 ∈ Prob) |
dstrvprob.2 | ⊢ (𝜑 → 𝑋 ∈ (rRndVar‘𝑃)) |
orvcelel.1 | ⊢ (𝜑 → 𝐴 ∈ 𝔅ℝ) |
Ref | Expression |
---|---|
orvcelval | ⊢ (𝜑 → (𝑋∘RV/𝑐 E 𝐴) = (◡𝑋 “ 𝐴)) |
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 | orrvcval4 31374 | . 2 ⊢ (𝜑 → (𝑋∘RV/𝑐 E 𝐴) = (◡𝑋 “ {𝑥 ∈ ℝ ∣ 𝑥 E 𝐴})) |
5 | epelg 5318 | . . . . . 6 ⊢ (𝐴 ∈ 𝔅ℝ → (𝑥 E 𝐴 ↔ 𝑥 ∈ 𝐴)) | |
6 | 3, 5 | syl 17 | . . . . 5 ⊢ (𝜑 → (𝑥 E 𝐴 ↔ 𝑥 ∈ 𝐴)) |
7 | 6 | rabbidv 3403 | . . . 4 ⊢ (𝜑 → {𝑥 ∈ ℝ ∣ 𝑥 E 𝐴} = {𝑥 ∈ ℝ ∣ 𝑥 ∈ 𝐴}) |
8 | dfin5 3837 | . . . . 5 ⊢ (ℝ ∩ 𝐴) = {𝑥 ∈ ℝ ∣ 𝑥 ∈ 𝐴} | |
9 | 8 | a1i 11 | . . . 4 ⊢ (𝜑 → (ℝ ∩ 𝐴) = {𝑥 ∈ ℝ ∣ 𝑥 ∈ 𝐴}) |
10 | elssuni 4741 | . . . . . . 7 ⊢ (𝐴 ∈ 𝔅ℝ → 𝐴 ⊆ ∪ 𝔅ℝ) | |
11 | unibrsiga 31096 | . . . . . . 7 ⊢ ∪ 𝔅ℝ = ℝ | |
12 | 10, 11 | syl6sseq 3907 | . . . . . 6 ⊢ (𝐴 ∈ 𝔅ℝ → 𝐴 ⊆ ℝ) |
13 | 3, 12 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐴 ⊆ ℝ) |
14 | sseqin2 4079 | . . . . 5 ⊢ (𝐴 ⊆ ℝ ↔ (ℝ ∩ 𝐴) = 𝐴) | |
15 | 13, 14 | sylib 210 | . . . 4 ⊢ (𝜑 → (ℝ ∩ 𝐴) = 𝐴) |
16 | 7, 9, 15 | 3eqtr2d 2820 | . . 3 ⊢ (𝜑 → {𝑥 ∈ ℝ ∣ 𝑥 E 𝐴} = 𝐴) |
17 | 16 | imaeq2d 5770 | . 2 ⊢ (𝜑 → (◡𝑋 “ {𝑥 ∈ ℝ ∣ 𝑥 E 𝐴}) = (◡𝑋 “ 𝐴)) |
18 | 4, 17 | eqtrd 2814 | 1 ⊢ (𝜑 → (𝑋∘RV/𝑐 E 𝐴) = (◡𝑋 “ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 198 = wceq 1507 ∈ wcel 2050 {crab 3092 ∩ cin 3828 ⊆ wss 3829 ∪ cuni 4712 class class class wbr 4929 E cep 5316 ◡ccnv 5406 “ cima 5410 ‘cfv 6188 (class class class)co 6976 ℝcr 10334 𝔅ℝcbrsiga 31091 Probcprb 31317 rRndVarcrrv 31350 ∘RV/𝑐corvc 31365 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-sep 5060 ax-nul 5067 ax-pow 5119 ax-pr 5186 ax-un 7279 ax-cnex 10391 ax-resscn 10392 ax-pre-lttri 10409 ax-pre-lttrn 10410 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-fal 1520 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ne 2968 df-nel 3074 df-ral 3093 df-rex 3094 df-rab 3097 df-v 3417 df-sbc 3682 df-csb 3787 df-dif 3832 df-un 3834 df-in 3836 df-ss 3843 df-nul 4179 df-if 4351 df-pw 4424 df-sn 4442 df-pr 4444 df-op 4448 df-uni 4713 df-int 4750 df-iun 4794 df-br 4930 df-opab 4992 df-mpt 5009 df-id 5312 df-eprel 5317 df-po 5326 df-so 5327 df-xp 5413 df-rel 5414 df-cnv 5415 df-co 5416 df-dm 5417 df-rn 5418 df-res 5419 df-ima 5420 df-iota 6152 df-fun 6190 df-fn 6191 df-f 6192 df-f1 6193 df-fo 6194 df-f1o 6195 df-fv 6196 df-ov 6979 df-oprab 6980 df-mpo 6981 df-1st 7501 df-2nd 7502 df-er 8089 df-map 8208 df-en 8307 df-dom 8308 df-sdom 8309 df-pnf 10476 df-mnf 10477 df-xr 10478 df-ltxr 10479 df-le 10480 df-ioo 12558 df-topgen 16573 df-top 21206 df-bases 21258 df-esum 30937 df-siga 31018 df-sigagen 31049 df-brsiga 31092 df-meas 31106 df-mbfm 31160 df-prob 31318 df-rrv 31351 df-orvc 31366 |
This theorem is referenced by: orvcelel 31379 dstrvval 31380 dstrvprob 31381 |
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