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 32143 | . 2 ⊢ (𝜑 → (𝑋∘RV/𝑐 E 𝐴) = (◡𝑋 “ {𝑥 ∈ ℝ ∣ 𝑥 E 𝐴})) |
5 | epelg 5461 | . . . . . 6 ⊢ (𝐴 ∈ 𝔅ℝ → (𝑥 E 𝐴 ↔ 𝑥 ∈ 𝐴)) | |
6 | 3, 5 | syl 17 | . . . . 5 ⊢ (𝜑 → (𝑥 E 𝐴 ↔ 𝑥 ∈ 𝐴)) |
7 | 6 | rabbidv 3390 | . . . 4 ⊢ (𝜑 → {𝑥 ∈ ℝ ∣ 𝑥 E 𝐴} = {𝑥 ∈ ℝ ∣ 𝑥 ∈ 𝐴}) |
8 | dfin5 3874 | . . . . 5 ⊢ (ℝ ∩ 𝐴) = {𝑥 ∈ ℝ ∣ 𝑥 ∈ 𝐴} | |
9 | 8 | a1i 11 | . . . 4 ⊢ (𝜑 → (ℝ ∩ 𝐴) = {𝑥 ∈ ℝ ∣ 𝑥 ∈ 𝐴}) |
10 | elssuni 4851 | . . . . . . 7 ⊢ (𝐴 ∈ 𝔅ℝ → 𝐴 ⊆ ∪ 𝔅ℝ) | |
11 | unibrsiga 31866 | . . . . . . 7 ⊢ ∪ 𝔅ℝ = ℝ | |
12 | 10, 11 | sseqtrdi 3951 | . . . . . 6 ⊢ (𝐴 ∈ 𝔅ℝ → 𝐴 ⊆ ℝ) |
13 | 3, 12 | syl 17 | . . . . 5 ⊢ (𝜑 → 𝐴 ⊆ ℝ) |
14 | sseqin2 4130 | . . . . 5 ⊢ (𝐴 ⊆ ℝ ↔ (ℝ ∩ 𝐴) = 𝐴) | |
15 | 13, 14 | sylib 221 | . . . 4 ⊢ (𝜑 → (ℝ ∩ 𝐴) = 𝐴) |
16 | 7, 9, 15 | 3eqtr2d 2783 | . . 3 ⊢ (𝜑 → {𝑥 ∈ ℝ ∣ 𝑥 E 𝐴} = 𝐴) |
17 | 16 | imaeq2d 5929 | . 2 ⊢ (𝜑 → (◡𝑋 “ {𝑥 ∈ ℝ ∣ 𝑥 E 𝐴}) = (◡𝑋 “ 𝐴)) |
18 | 4, 17 | eqtrd 2777 | 1 ⊢ (𝜑 → (𝑋∘RV/𝑐 E 𝐴) = (◡𝑋 “ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 = wceq 1543 ∈ wcel 2110 {crab 3065 ∩ cin 3865 ⊆ wss 3866 ∪ cuni 4819 class class class wbr 5053 E cep 5459 ◡ccnv 5550 “ cima 5554 ‘cfv 6380 (class class class)co 7213 ℝcr 10728 𝔅ℝcbrsiga 31861 Probcprb 32086 rRndVarcrrv 32119 ∘RV/𝑐corvc 32134 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pow 5258 ax-pr 5322 ax-un 7523 ax-cnex 10785 ax-resscn 10786 ax-pre-lttri 10803 ax-pre-lttrn 10804 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-nel 3047 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-int 4860 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-id 5455 df-eprel 5460 df-po 5468 df-so 5469 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-ov 7216 df-oprab 7217 df-mpo 7218 df-1st 7761 df-2nd 7762 df-er 8391 df-map 8510 df-en 8627 df-dom 8628 df-sdom 8629 df-pnf 10869 df-mnf 10870 df-xr 10871 df-ltxr 10872 df-le 10873 df-ioo 12939 df-topgen 16948 df-top 21791 df-bases 21843 df-esum 31708 df-siga 31789 df-sigagen 31819 df-brsiga 31862 df-meas 31876 df-mbfm 31930 df-prob 32087 df-rrv 32120 df-orvc 32135 |
This theorem is referenced by: orvcelel 32148 dstrvval 32149 dstrvprob 32150 |
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