This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

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 ) ) } )