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Definition df-orvc 26976
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 5093- elementhood:  { X  e.  A } as  ( XRV/𝑐  _E  A ) cf. epel 4736- less-than:  { X  <_  A } as  ( XRV/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/𝑐 R  =  ( x  e.  { x  |  Fun  x } , 
a  e.  _V  |->  ( `' x " { y  |  y R a } ) )
Distinct variable group:    x, a, y, R

Detailed syntax breakdown of Definition df-orvc
StepHypRef Expression
1 cR . . 3  class  R
21corvc 26975 . 2  classRV/𝑐 R
3 vx . . 3  setvar  x
4 va . . 3  setvar  a
53cv 1369 . . . . 5  class  x
65wfun 5513 . . . 4  wff  Fun  x
76, 3cab 2436 . . 3  class  { x  |  Fun  x }
8 cvv 3071 . . 3  class  _V
95ccnv 4940 . . . 4  class  `' x
10 vy . . . . . . 7  setvar  y
1110cv 1369 . . . . . 6  class  y
124cv 1369 . . . . . 6  class  a
1311, 12, 1wbr 4393 . . . . 5  wff  y R a
1413, 10cab 2436 . . . 4  class  { y  |  y R a }
159, 14cima 4944 . . 3  class  ( `' x " { y  |  y R a } )
163, 4, 7, 8, 15cmpt2 6195 . 2  class  ( x  e.  { x  |  Fun  x } , 
a  e.  _V  |->  ( `' x " { y  |  y R a } ) )
172, 16wceq 1370 1  wffRV/𝑐 R  =  ( x  e.  { x  |  Fun  x } , 
a  e.  _V  |->  ( `' x " { y  |  y R a } ) )
Colors of variables: wff setvar class
This definition is referenced by:  orvcval  26977
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