Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > orvcval | Structured version Visualization version GIF version |
Description: Value of the preimage mapping operator applied on a given random variable and constant. (Contributed by Thierry Arnoux, 19-Jan-2017.) |
Ref | Expression |
---|---|
orvcval.1 | ⊢ (𝜑 → Fun 𝑋) |
orvcval.2 | ⊢ (𝜑 → 𝑋 ∈ 𝑉) |
orvcval.3 | ⊢ (𝜑 → 𝐴 ∈ 𝑊) |
Ref | Expression |
---|---|
orvcval | ⊢ (𝜑 → (𝑋∘RV/𝑐𝑅𝐴) = (◡𝑋 “ {𝑦 ∣ 𝑦𝑅𝐴})) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-orvc 31613 | . . 3 ⊢ ∘RV/𝑐𝑅 = (𝑥 ∈ {𝑥 ∣ Fun 𝑥}, 𝑎 ∈ V ↦ (◡𝑥 “ {𝑦 ∣ 𝑦𝑅𝑎})) | |
2 | 1 | a1i 11 | . 2 ⊢ (𝜑 → ∘RV/𝑐𝑅 = (𝑥 ∈ {𝑥 ∣ Fun 𝑥}, 𝑎 ∈ V ↦ (◡𝑥 “ {𝑦 ∣ 𝑦𝑅𝑎}))) |
3 | simpl 483 | . . . . 5 ⊢ ((𝑥 = 𝑋 ∧ 𝑎 = 𝐴) → 𝑥 = 𝑋) | |
4 | 3 | cnveqd 5739 | . . . 4 ⊢ ((𝑥 = 𝑋 ∧ 𝑎 = 𝐴) → ◡𝑥 = ◡𝑋) |
5 | simpr 485 | . . . . . 6 ⊢ ((𝑥 = 𝑋 ∧ 𝑎 = 𝐴) → 𝑎 = 𝐴) | |
6 | 5 | breq2d 5069 | . . . . 5 ⊢ ((𝑥 = 𝑋 ∧ 𝑎 = 𝐴) → (𝑦𝑅𝑎 ↔ 𝑦𝑅𝐴)) |
7 | 6 | abbidv 2882 | . . . 4 ⊢ ((𝑥 = 𝑋 ∧ 𝑎 = 𝐴) → {𝑦 ∣ 𝑦𝑅𝑎} = {𝑦 ∣ 𝑦𝑅𝐴}) |
8 | 4, 7 | imaeq12d 5923 | . . 3 ⊢ ((𝑥 = 𝑋 ∧ 𝑎 = 𝐴) → (◡𝑥 “ {𝑦 ∣ 𝑦𝑅𝑎}) = (◡𝑋 “ {𝑦 ∣ 𝑦𝑅𝐴})) |
9 | 8 | adantl 482 | . 2 ⊢ ((𝜑 ∧ (𝑥 = 𝑋 ∧ 𝑎 = 𝐴)) → (◡𝑥 “ {𝑦 ∣ 𝑦𝑅𝑎}) = (◡𝑋 “ {𝑦 ∣ 𝑦𝑅𝐴})) |
10 | orvcval.2 | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝑉) | |
11 | orvcval.1 | . . 3 ⊢ (𝜑 → Fun 𝑋) | |
12 | funeq 6368 | . . 3 ⊢ (𝑥 = 𝑋 → (Fun 𝑥 ↔ Fun 𝑋)) | |
13 | 10, 11, 12 | elabd 3666 | . 2 ⊢ (𝜑 → 𝑋 ∈ {𝑥 ∣ Fun 𝑥}) |
14 | orvcval.3 | . . 3 ⊢ (𝜑 → 𝐴 ∈ 𝑊) | |
15 | elex 3510 | . . 3 ⊢ (𝐴 ∈ 𝑊 → 𝐴 ∈ V) | |
16 | 14, 15 | syl 17 | . 2 ⊢ (𝜑 → 𝐴 ∈ V) |
17 | cnvexg 7618 | . . 3 ⊢ (𝑋 ∈ 𝑉 → ◡𝑋 ∈ V) | |
18 | imaexg 7609 | . . 3 ⊢ (◡𝑋 ∈ V → (◡𝑋 “ {𝑦 ∣ 𝑦𝑅𝐴}) ∈ V) | |
19 | 10, 17, 18 | 3syl 18 | . 2 ⊢ (𝜑 → (◡𝑋 “ {𝑦 ∣ 𝑦𝑅𝐴}) ∈ V) |
20 | 2, 9, 13, 16, 19 | ovmpod 7291 | 1 ⊢ (𝜑 → (𝑋∘RV/𝑐𝑅𝐴) = (◡𝑋 “ {𝑦 ∣ 𝑦𝑅𝐴})) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 {cab 2796 Vcvv 3492 class class class wbr 5057 ◡ccnv 5547 “ cima 5551 Fun wfun 6342 (class class class)co 7145 ∈ cmpo 7147 ∘RV/𝑐corvc 31612 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-orvc 31613 |
This theorem is referenced by: orvcval2 31615 orvcval4 31617 |
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