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Definition df-orvc 30849
Description: Define the preimage set mapping operator. In probability theory, the notation 𝑃(𝑋 = 𝐴) denotes the probability that a random variable 𝑋 takes the value 𝐴. We introduce here an operator which enables to write this in Metamath as (𝑃‘(𝑋RV/𝑐 I 𝐴)), and keep a similar notation. Because with this notation (𝑋RV/𝑐 I 𝐴) is a set, we can also apply it to conditional probabilities, like in (𝑃‘(𝑋RV/𝑐 I 𝐴) ∣ (𝑌RV/𝑐 I 𝐵))).

The oRVC operator transforms a relation 𝑅 into an operation taking a random variable 𝑋 and a constant 𝐶, and returning the preimage through 𝑋 of the equivalence class of 𝐶.

The most commonly used relations are: - equality: {𝑋 = 𝐴} as (𝑋RV/𝑐 I 𝐴) cf. ideq 5431- elementhood: {𝑋𝐴} as (𝑋RV/𝑐 E 𝐴) cf. epel 5183- less-than: {𝑋𝐴} as (𝑋RV/𝑐𝐴)

Even though it is primarily designed to be used within probability theory and with random variables, this operator is defined on generic functions, and could be used in other fields, e.g. for continuous functions. (Contributed by Thierry Arnoux, 15-Jan-2017.)

Assertion
Ref Expression
df-orvc RV/𝑐𝑅 = (𝑥 ∈ {𝑥 ∣ Fun 𝑥}, 𝑎 ∈ V ↦ (𝑥 “ {𝑦𝑦𝑅𝑎}))
Distinct variable group:   𝑥,𝑎,𝑦,𝑅

Detailed syntax breakdown of Definition df-orvc
StepHypRef Expression
1 cR . . 3 class 𝑅
21corvc 30848 . 2 class RV/𝑐𝑅
3 vx . . 3 setvar 𝑥
4 va . . 3 setvar 𝑎
53cv 1631 . . . . 5 class 𝑥
65wfun 6044 . . . 4 wff Fun 𝑥
76, 3cab 2747 . . 3 class {𝑥 ∣ Fun 𝑥}
8 cvv 3341 . . 3 class V
95ccnv 5266 . . . 4 class 𝑥
10 vy . . . . . . 7 setvar 𝑦
1110cv 1631 . . . . . 6 class 𝑦
124cv 1631 . . . . . 6 class 𝑎
1311, 12, 1wbr 4805 . . . . 5 wff 𝑦𝑅𝑎
1413, 10cab 2747 . . . 4 class {𝑦𝑦𝑅𝑎}
159, 14cima 5270 . . 3 class (𝑥 “ {𝑦𝑦𝑅𝑎})
163, 4, 7, 8, 15cmpt2 6817 . 2 class (𝑥 ∈ {𝑥 ∣ Fun 𝑥}, 𝑎 ∈ V ↦ (𝑥 “ {𝑦𝑦𝑅𝑎}))
172, 16wceq 1632 1 wff RV/𝑐𝑅 = (𝑥 ∈ {𝑥 ∣ Fun 𝑥}, 𝑎 ∈ V ↦ (𝑥 “ {𝑦𝑦𝑅𝑎}))
Colors of variables: wff setvar class
This definition is referenced by:  orvcval  30850
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