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Definition df-orvc 30825
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 5428- elementhood: {𝑋𝐴} as (𝑋RV/𝑐 E 𝐴) cf. epel 5180- 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 30824 . 2 class RV/𝑐𝑅
3 vx . . 3 setvar 𝑥
4 va . . 3 setvar 𝑎
53cv 1629 . . . . 5 class 𝑥
65wfun 6041 . . . 4 wff Fun 𝑥
76, 3cab 2744 . . 3 class {𝑥 ∣ Fun 𝑥}
8 cvv 3338 . . 3 class V
95ccnv 5263 . . . 4 class 𝑥
10 vy . . . . . . 7 setvar 𝑦
1110cv 1629 . . . . . 6 class 𝑦
124cv 1629 . . . . . 6 class 𝑎
1311, 12, 1wbr 4802 . . . . 5 wff 𝑦𝑅𝑎
1413, 10cab 2744 . . . 4 class {𝑦𝑦𝑅𝑎}
159, 14cima 5267 . . 3 class (𝑥 “ {𝑦𝑦𝑅𝑎})
163, 4, 7, 8, 15cmpt2 6813 . 2 class (𝑥 ∈ {𝑥 ∣ Fun 𝑥}, 𝑎 ∈ V ↦ (𝑥 “ {𝑦𝑦𝑅𝑎}))
172, 16wceq 1630 1 wff RV/𝑐𝑅 = (𝑥 ∈ {𝑥 ∣ Fun 𝑥}, 𝑎 ∈ V ↦ (𝑥 “ {𝑦𝑦𝑅𝑎}))
Colors of variables: wff setvar class
This definition is referenced by:  orvcval  30826
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