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Theorem cfval 8083
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 8082 . . 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 4318 . . 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 189 . 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 3330 . . . . . . . 8  |-  ( v  =  A  ->  (
y  C_  v  <->  y  C_  A ) )
5 raleq 2864 . . . . . . . 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 692 . . . . . . 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 685 . . . . . 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 1633 . . . . 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 2518 . . . 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 4015 . . 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 7784 . . 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 5763 . 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 650 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 359   E.wex 1547    = wceq 1649    e. wcel 1721   {cab 2390   A.wral 2666   E.wrex 2667   _Vcvv 2916    C_ wss 3280   |^|cint 4010   Oncon0 4541   ` cfv 5413   cardccrd 7778   cfccf 7780
This theorem is referenced by:  cfub  8085  cflm  8086  cardcf  8088  cflecard  8089  cfeq0  8092  cfsuc  8093  cff1  8094  cfflb  8095  cfval2  8096  cflim3  8098
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-int 4011  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-iota 5377  df-fun 5415  df-fv 5421  df-cf 7784
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