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Definition df-rank 7370
Description: Define the rank function. See rankval 7421, rankval2 7423, rankval3 7445, or rankval4 7472 its value. The rank is a kind of "inverse" of the cumulative hierarchy of sets function 
R1: given a set, it returns an ordinal number telling us the smallest layer of the hierarchy to which the set belongs. Based on Definition 9.14 of [TakeutiZaring] p. 79. Theorem rankid 7438 illustrates the "inverse" concept. Another nice theorem showing the relationship is rankr1a 7441. (Contributed by NM, 11-Oct-2003.)
Assertion
Ref Expression
df-rank  |-  rank  =  ( x  e.  _V  |->  |^|
{ y  e.  On  |  x  e.  ( R1 `  suc  y ) } )
Distinct variable group:    x, y

Detailed syntax breakdown of Definition df-rank
StepHypRef Expression
1 crnk 7368 . 2  class  rank
2 vx . . 3  set  x
3 cvv 2740 . . 3  class  _V
42cv 1618 . . . . . 6  class  x
5 vy . . . . . . . . 9  set  y
65cv 1618 . . . . . . . 8  class  y
76csuc 4331 . . . . . . 7  class  suc  y
8 cr1 7367 . . . . . . 7  class  R1
97, 8cfv 4638 . . . . . 6  class  ( R1
`  suc  y )
104, 9wcel 1621 . . . . 5  wff  x  e.  ( R1 `  suc  y )
11 con0 4329 . . . . 5  class  On
1210, 5, 11crab 2519 . . . 4  class  { y  e.  On  |  x  e.  ( R1 `  suc  y ) }
1312cint 3803 . . 3  class  |^| { y  e.  On  |  x  e.  ( R1 `  suc  y ) }
142, 3, 13cmpt 4017 . 2  class  ( x  e.  _V  |->  |^| { y  e.  On  |  x  e.  ( R1 `  suc  y ) } )
151, 14wceq 1619 1  wff  rank  =  ( x  e.  _V  |->  |^|
{ y  e.  On  |  x  e.  ( R1 `  suc  y ) } )
Colors of variables: wff set class
This definition is referenced by:  rankf  7399  rankvalb  7402
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