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Theorem cfval 7889
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 7888 . . 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 4185 . . 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 3213 . . . . . . . 8  |-  ( v  =  A  ->  (
y  C_  v  <->  y  C_  A ) )
5 raleq 2749 . . . . . . . 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 1616 . . . . 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 2410 . . . 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 3883 . . 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 7590 . . 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 5616 . 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 1531    = wceq 1632    e. wcel 1696   {cab 2282   A.wral 2556   E.wrex 2557   _Vcvv 2801    C_ wss 3165   |^|cint 3878   Oncon0 4408   ` cfv 5271   cardccrd 7584   cfccf 7586
This theorem is referenced by:  cfub  7891  cflm  7892  cardcf  7894  cflecard  7895  cfeq0  7898  cfsuc  7899  cff1  7900  cfflb  7901  cfval2  7902  cflim3  7904
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-int 3879  df-br 4040  df-opab 4094  df-mpt 4095  df-id 4325  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-iota 5235  df-fun 5273  df-fv 5279  df-cf 7590
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