Definition df-rank 8114

Description: Define the rank function. See rankval 8165, rankval2 8167, rankval3 8189, or rankval4 8216 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 8182 illustrates the "inverse" concept. Another nice theorem showing the relationship is rankr1a 8185. (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

Step	Hyp	Ref	Expression
1		crnk 8112	. 2 class rank
2		vx	. . 3 setvar x
3		cvv 3047	. . 3 class _V
4	2	cv 1398	. . . . . 6 class x
5		vy	. . . . . . . . 9 setvar y
6	5	cv 1398	. . . . . . . 8 class y
7	6	csuc 4807	. . . . . . 7 class suc y
8		cr1 8111	. . . . . . 7 class R1
9	7, 8	cfv 5509	. . . . . 6 class ( R1 ` suc y )
10	4, 9	wcel 1836	. . . . 5 wff x e. ( R1 ` suc y )
11		con0 4805	. . . . 5 class On
12	10, 5, 11	crab 2746	. . . 4 class { y e. On | x e. ( R1 ` suc y ) }
13	12	cint 4212	. . 3 class |^| { y e. On | x e. ( R1 ` suc y ) }
14	2, 3, 13	cmpt 4438	. 2 class ( x e. _V |-> |^| { y e. On | x e. ( R1 ` suc y ) } )
15	1, 14	wceq 1399	1 wff rank = ( x e. _V |-> |^| { y e. On | x e. ( R1 ` suc y ) } )

This definition is referenced by:  rankf  8143  rankvalb  8146