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Theorem cflim3 7904
Description: Another expression for the cofinality function. (Contributed by Mario Carneiro, 28-Feb-2013.)
Hypothesis
Ref Expression
cflim3.1  |-  A  e. 
_V
Assertion
Ref Expression
cflim3  |-  ( Lim 
A  ->  ( cf `  A )  =  |^|_ x  e.  { x  e. 
~P A  |  U. x  =  A } 
( card `  x )
)
Distinct variable group:    x, A

Proof of Theorem cflim3
Dummy variables  w  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 limord 4467 . . . 4  |-  ( Lim 
A  ->  Ord  A )
2 cflim3.1 . . . . 5  |-  A  e. 
_V
32elon 4417 . . . 4  |-  ( A  e.  On  <->  Ord  A )
41, 3sylibr 203 . . 3  |-  ( Lim 
A  ->  A  e.  On )
5 cfval 7889 . . 3  |-  ( A  e.  On  ->  ( cf `  A )  = 
|^| { y  |  E. x ( y  =  ( card `  x
)  /\  ( x  C_  A  /\  A. z  e.  A  E. w  e.  x  z  C_  w ) ) } )
64, 5syl 15 . 2  |-  ( Lim 
A  ->  ( cf `  A )  =  |^| { y  |  E. x
( y  =  (
card `  x )  /\  ( x  C_  A  /\  A. z  e.  A  E. w  e.  x  z  C_  w ) ) } )
7 fvex 5555 . . . 4  |-  ( card `  x )  e.  _V
87dfiin2 3954 . . 3  |-  |^|_ x  e.  { x  e.  ~P A  |  U. x  =  A }  ( card `  x )  =  |^| { y  |  E. x  e.  { x  e.  ~P A  |  U. x  =  A } y  =  ( card `  x
) }
9 df-rex 2562 . . . . . 6  |-  ( E. x  e.  { x  e.  ~P A  |  U. x  =  A }
y  =  ( card `  x )  <->  E. x
( x  e.  {
x  e.  ~P A  |  U. x  =  A }  /\  y  =  ( card `  x
) ) )
10 ancom 437 . . . . . . . 8  |-  ( ( x  e.  { x  e.  ~P A  |  U. x  =  A }  /\  y  =  ( card `  x ) )  <-> 
( y  =  (
card `  x )  /\  x  e.  { x  e.  ~P A  |  U. x  =  A }
) )
11 rabid 2729 . . . . . . . . . 10  |-  ( x  e.  { x  e. 
~P A  |  U. x  =  A }  <->  ( x  e.  ~P A  /\  U. x  =  A ) )
122elpw2 4191 . . . . . . . . . . . 12  |-  ( x  e.  ~P A  <->  x  C_  A
)
1312anbi1i 676 . . . . . . . . . . 11  |-  ( ( x  e.  ~P A  /\  U. x  =  A )  <->  ( x  C_  A  /\  U. x  =  A ) )
14 coflim 7903 . . . . . . . . . . . 12  |-  ( ( Lim  A  /\  x  C_  A )  ->  ( U. x  =  A  <->  A. z  e.  A  E. w  e.  x  z  C_  w ) )
1514pm5.32da 622 . . . . . . . . . . 11  |-  ( Lim 
A  ->  ( (
x  C_  A  /\  U. x  =  A )  <-> 
( x  C_  A  /\  A. z  e.  A  E. w  e.  x  z  C_  w ) ) )
1613, 15syl5bb 248 . . . . . . . . . 10  |-  ( Lim 
A  ->  ( (
x  e.  ~P A  /\  U. x  =  A )  <->  ( x  C_  A  /\  A. z  e.  A  E. w  e.  x  z  C_  w
) ) )
1711, 16syl5bb 248 . . . . . . . . 9  |-  ( Lim 
A  ->  ( x  e.  { x  e.  ~P A  |  U. x  =  A }  <->  ( x  C_  A  /\  A. z  e.  A  E. w  e.  x  z  C_  w ) ) )
1817anbi2d 684 . . . . . . . 8  |-  ( Lim 
A  ->  ( (
y  =  ( card `  x )  /\  x  e.  { x  e.  ~P A  |  U. x  =  A } )  <->  ( y  =  ( card `  x
)  /\  ( x  C_  A  /\  A. z  e.  A  E. w  e.  x  z  C_  w ) ) ) )
1910, 18syl5bb 248 . . . . . . 7  |-  ( Lim 
A  ->  ( (
x  e.  { x  e.  ~P A  |  U. x  =  A }  /\  y  =  ( card `  x ) )  <-> 
( y  =  (
card `  x )  /\  ( x  C_  A  /\  A. z  e.  A  E. w  e.  x  z  C_  w ) ) ) )
2019exbidv 1616 . . . . . 6  |-  ( Lim 
A  ->  ( E. x ( x  e. 
{ x  e.  ~P A  |  U. x  =  A }  /\  y  =  ( card `  x
) )  <->  E. x
( y  =  (
card `  x )  /\  ( x  C_  A  /\  A. z  e.  A  E. w  e.  x  z  C_  w ) ) ) )
219, 20syl5bb 248 . . . . 5  |-  ( Lim 
A  ->  ( E. x  e.  { x  e.  ~P A  |  U. x  =  A }
y  =  ( card `  x )  <->  E. x
( y  =  (
card `  x )  /\  ( x  C_  A  /\  A. z  e.  A  E. w  e.  x  z  C_  w ) ) ) )
2221abbidv 2410 . . . 4  |-  ( Lim 
A  ->  { y  |  E. x  e.  {
x  e.  ~P A  |  U. x  =  A } y  =  (
card `  x ) }  =  { y  |  E. x ( y  =  ( card `  x
)  /\  ( x  C_  A  /\  A. z  e.  A  E. w  e.  x  z  C_  w ) ) } )
2322inteqd 3883 . . 3  |-  ( Lim 
A  ->  |^| { y  |  E. x  e. 
{ x  e.  ~P A  |  U. x  =  A } y  =  ( card `  x
) }  =  |^| { y  |  E. x
( y  =  (
card `  x )  /\  ( x  C_  A  /\  A. z  e.  A  E. w  e.  x  z  C_  w ) ) } )
248, 23syl5req 2341 . 2  |-  ( Lim 
A  ->  |^| { y  |  E. x ( y  =  ( card `  x )  /\  (
x  C_  A  /\  A. z  e.  A  E. w  e.  x  z  C_  w ) ) }  =  |^|_ x  e.  {
x  e.  ~P A  |  U. x  =  A }  ( card `  x
) )
256, 24eqtrd 2328 1  |-  ( Lim 
A  ->  ( cf `  A )  =  |^|_ x  e.  { x  e. 
~P A  |  U. x  =  A } 
( card `  x )
)
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   {crab 2560   _Vcvv 2801    C_ wss 3165   ~Pcpw 3638   U.cuni 3843   |^|cint 3878   |^|_ciin 3922   Ord word 4407   Oncon0 4408   Lim wlim 4409   ` cfv 5271   cardccrd 7584   cfccf 7586
This theorem is referenced by:  cflim2  7905  cfss  7907  cfslb  7908
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-pw 3640  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-int 3879  df-iin 3924  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  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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