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Theorem cfval 8129
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.)
Assertion
Ref Expression
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 ) ) } )
Distinct variable group:    x, y, z, w, A

Proof of Theorem cfval
Dummy variable  v is distinct from all other variables.
StepHypRef Expression
1 cflem 8128 . . 3  |-  ( 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 4360 . . 3  |-  ( 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 )
31, 2sylib 190 . 2  |-  ( 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 3372 . . . . . . . 8  |-  ( v  =  A  ->  (
y  C_  v  <->  y  C_  A ) )
5 raleq 2906 . . . . . . . 8  |-  ( v  =  A  ->  ( A. z  e.  v  E. w  e.  y 
z  C_  w  <->  A. z  e.  A  E. w  e.  y  z  C_  w ) )
64, 5anbi12d 693 . . . . . . 7  |-  ( 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 ) ) )
76anbi2d 686 . . . . . 6  |-  ( 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 ) ) ) )
87exbidv 1637 . . . . 5  |-  ( 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 )
) ) )
98abbidv 2552 . . . 4  |-  ( 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 ) ) } )
109inteqd 4057 . . 3  |-  ( 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 7830 . . 3  |-  cf  =  ( v  e.  On  |->  |^|
{ x  |  E. y ( x  =  ( card `  y
)  /\  ( y  C_  v  /\  A. z  e.  v  E. w  e.  y  z  C_  w ) ) } )
1210, 11fvmptg 5806 . 2  |-  ( ( 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 ) ) } )
133, 12mpdan 651 1  |-  ( 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 ) ) } )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360   E.wex 1551    = wceq 1653    e. wcel 1726   {cab 2424   A.wral 2707   E.wrex 2708   _Vcvv 2958    C_ wss 3322   |^|cint 4052   Oncon0 4583   ` cfv 5456   cardccrd 7824   cfccf 7826
This theorem is referenced by:  cfub  8131  cflm  8132  cardcf  8134  cflecard  8135  cfeq0  8138  cfsuc  8139  cff1  8140  cfflb  8141  cfval2  8142  cflim3  8144
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-int 4053  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-iota 5420  df-fun 5458  df-fv 5464  df-cf 7830
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