Definition df-orvc 34764

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‘(𝑋∘_RV/𝑐 I 𝐴)), and keep a similar notation. Because with this notation (𝑋∘_RV/𝑐 I 𝐴) is a set, we can also apply it to conditional probabilities, like in (𝑃‘(𝑋∘_RV/𝑐 I 𝐴) ∣ (𝑌∘_RV/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 (𝑋∘_RV/𝑐 I 𝐴) cf. ideq 5829- elementhood: {𝑋 ∈ 𝐴} as (𝑋∘_RV/𝑐 E 𝐴) cf. epel 5555- less-than: {𝑋 ≤ 𝐴} as (𝑋∘_RV/𝑐 ≤ 𝐴)

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 “ {𝑦 ∣ 𝑦𝑅𝑎}))

Distinct variable group:   𝑥,𝑎,𝑦,𝑅

Detailed syntax breakdown of Definition df-orvc

Step	Hyp	Ref	Expression
1		cR	. . 3 class 𝑅
2	1	corvc 34763	. 2 class ∘RV/𝑐𝑅
3		vx	. . 3 setvar 𝑥
4		va	. . 3 setvar 𝑎
5	3	cv 1562	. . . . 5 class 𝑥
6	5	wfun 6519	. . . 4 wff Fun 𝑥
7	6, 3	cab 2743	. . 3 class {𝑥 ∣ Fun 𝑥}
8		cvv 3457	. . 3 class V
9	5	ccnv 5651	. . . 4 class ◡𝑥
10		vy	. . . . . . 7 setvar 𝑦
11	10	cv 1562	. . . . . 6 class 𝑦
12	4	cv 1562	. . . . . 6 class 𝑎
13	11, 12, 1	wbr 5105	. . . . 5 wff 𝑦𝑅𝑎
14	13, 10	cab 2743	. . . 4 class {𝑦 ∣ 𝑦𝑅𝑎}
15	9, 14	cima 5655	. . 3 class (◡𝑥 “ {𝑦 ∣ 𝑦𝑅𝑎})
16	3, 4, 7, 8, 15	cmpo 7402	. 2 class (𝑥 ∈ {𝑥 ∣ Fun 𝑥}, 𝑎 ∈ V ↦ (◡𝑥 “ {𝑦 ∣ 𝑦𝑅𝑎}))
17	2, 16	wceq 1563	1 wff ∘RV/𝑐𝑅 = (𝑥 ∈ {𝑥 ∣ Fun 𝑥}, 𝑎 ∈ V ↦ (◡𝑥 “ {𝑦 ∣ 𝑦𝑅𝑎}))

This definition is referenced by:  orvcval  34765