Definition df-rank 8243

Description: Define the rank function. See rankval 8294, rankval2 8296, rankval3 8318, or rankval4 8345 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 8311 illustrates the "inverse" concept. Another nice theorem showing the relationship is rankr1a 8314. (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 8241	. 2 class rank
2		vx	. . 3 setvar x
3		cvv 3082	. . 3 class _V
4	2	cv 1437	. . . . . 6 class x
5		vy	. . . . . . . . 9 setvar y
6	5	cv 1437	. . . . . . . 8 class y
7	6	csuc 5443	. . . . . . 7 class suc y
8		cr1 8240	. . . . . . 7 class R1
9	7, 8	cfv 5600	. . . . . 6 class ( R1 ` suc y )
10	4, 9	wcel 1869	. . . . 5 wff x e. ( R1 ` suc y )
11		con0 5441	. . . . 5 class On
12	10, 5, 11	crab 2780	. . . 4 class { y e. On | x e. ( R1 ` suc y ) }
13	12	cint 4254	. . 3 class |^| { y e. On | x e. ( R1 ` suc y ) }
14	2, 3, 13	cmpt 4481	. 2 class ( x e. _V |-> |^| { y e. On | x e. ( R1 ` suc y ) } )
15	1, 14	wceq 1438	1 wff rank = ( x e. _V |-> |^| { y e. On | x e. ( R1 ` suc y ) } )

This definition is referenced by:  rankf  8272  rankvalb  8275