Theorem cfval

Description: Value of the cofinality function. Definition B of Saharon Shelah, Cardinal Arithmetic (1994), p. xxx (Roman numeral 30). The cofinality of an ordinal number A is the cardinality (size) of the smallest unbounded subset y of the ordinal number. Unbounded means that for every member of A , there is a member of y that is at least as large. Cofinality is a measure of how "reachable from below" an ordinal is. (Contributed by NM, 1-Apr-2004) (Revised by Mario Carneiro, 15-Sep-2013)

	Ref	Expression
Assertion	cfval	|- ( A e. On -> ( cf ` A ) = |^| { x | E. y ( x = ( card ` y ) /\ ( y C_ A /\ A. z e. A E. w e. y z C_ w ) ) } )

Proof

Step	Hyp	Ref	Expression
1		cflem	|- ( A e. On -> E. x E. y ( x = ( card ` y ) /\ ( y C_ A /\ A. z e. A E. w e. y z C_ w ) ) )
2		intexab	|- ( E. x E. y ( x = ( card ` y ) /\ ( y C_ A /\ A. z e. A E. w e. y z C_ w ) ) <-> |^| { x | E. y ( x = ( card ` y ) /\ ( y C_ A /\ A. z e. A E. w e. y z C_ w ) ) } e. _V )
3	1 2	sylib	|- ( A e. On -> |^| { x | E. y ( x = ( card ` y ) /\ ( y C_ A /\ A. z e. A E. w e. y z C_ w ) ) } e. _V )
4		sseq2	|- ( v = A -> ( y C_ v <-> y C_ A ) )
5		raleq	|- ( v = A -> ( A. z e. v E. w e. y z C_ w <-> A. z e. A E. w e. y z C_ w ) )
6	4 5	anbi12d	|- ( v = A -> ( ( y C_ v /\ A. z e. v E. w e. y z C_ w ) <-> ( y C_ A /\ A. z e. A E. w e. y z C_ w ) ) )
7	6	anbi2d	|- ( v = A -> ( ( x = ( card ` y ) /\ ( y C_ v /\ A. z e. v E. w e. y z C_ w ) ) <-> ( x = ( card ` y ) /\ ( y C_ A /\ A. z e. A E. w e. y z C_ w ) ) ) )
8	7	exbidv	|- ( v = A -> ( E. y ( x = ( card ` y ) /\ ( y C_ v /\ A. z e. v E. w e. y z C_ w ) ) <-> E. y ( x = ( card ` y ) /\ ( y C_ A /\ A. z e. A E. w e. y z C_ w ) ) ) )
9	8	abbidv	|- ( v = A -> { x | E. y ( x = ( card ` y ) /\ ( y C_ v /\ A. z e. v E. w e. y z C_ w ) ) } = { x | E. y ( x = ( card ` y ) /\ ( y C_ A /\ A. z e. A E. w e. y z C_ w ) ) } )
10	9	inteqd	|- ( v = A -> |^| { x | E. y ( x = ( card ` y ) /\ ( y C_ v /\ A. z e. v E. w e. y z C_ w ) ) } = |^| { x | E. y ( x = ( card ` y ) /\ ( y C_ A /\ A. z e. A E. w e. y z C_ w ) ) } )
11		df-cf	|- cf = ( v e. On |-> |^| { x | E. y ( x = ( card ` y ) /\ ( y C_ v /\ A. z e. v E. w e. y z C_ w ) ) } )
12	10 11	fvmptg	|- ( ( A e. On /\ |^| { x | E. y ( x = ( card ` y ) /\ ( y C_ A /\ A. z e. A E. w e. y z C_ w ) ) } e. _V ) -> ( cf ` A ) = |^| { x | E. y ( x = ( card ` y ) /\ ( y C_ A /\ A. z e. A E. w e. y z C_ w ) ) } )
13	3 12	mpdan	|- ( A e. On -> ( cf ` A ) = |^| { x | E. y ( x = ( card ` y ) /\ ( y C_ A /\ A. z e. A E. w e. y z C_ w ) ) } )