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Theorem cfval 7873
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 7872 . . 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 4169 . . 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 188 . 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 3200 . . . . . . . 8  |-  ( v  =  A  ->  (
y  C_  v  <->  y  C_  A ) )
5 raleq 2736 . . . . . . . 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 691 . . . . . . 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 684 . . . . . 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 1612 . . . . 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 2397 . . . 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 3867 . . 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 7574 . . 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 5600 . 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 649 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 358   E.wex 1528    = wceq 1623    e. wcel 1684   {cab 2269   A.wral 2543   E.wrex 2544   _Vcvv 2788    C_ wss 3152   |^|cint 3862   Oncon0 4392   ` cfv 5255   cardccrd 7568   cfccf 7570
This theorem is referenced by:  cfub  7875  cflm  7876  cardcf  7878  cflecard  7879  cfeq0  7882  cfsuc  7883  cff1  7884  cfflb  7885  cfval2  7886  cflim3  7888
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-int 3863  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-iota 5219  df-fun 5257  df-fv 5263  df-cf 7574
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