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Theorem cflim3 7888
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 4451 . . . 4  |-  ( Lim 
A  ->  Ord  A )
2 cflim3.1 . . . . 5  |-  A  e. 
_V
32elon 4401 . . . 4  |-  ( A  e.  On  <->  Ord  A )
41, 3sylibr 203 . . 3  |-  ( Lim 
A  ->  A  e.  On )
5 cfval 7873 . . 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 5539 . . . 4  |-  ( card `  x )  e.  _V
87dfiin2 3938 . . 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 2549 . . . . . 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 2716 . . . . . . . . . 10  |-  ( x  e.  { x  e. 
~P A  |  U. x  =  A }  <->  ( x  e.  ~P A  /\  U. x  =  A ) )
122elpw2 4175 . . . . . . . . . . . 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 7887 . . . . . . . . . . . 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 1612 . . . . . 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 2397 . . . 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 3867 . . 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 2328 . 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 2315 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 1528    = wceq 1623    e. wcel 1684   {cab 2269   A.wral 2543   E.wrex 2544   {crab 2547   _Vcvv 2788    C_ wss 3152   ~Pcpw 3625   U.cuni 3827   |^|cint 3862   |^|_ciin 3906   Ord word 4391   Oncon0 4392   Lim wlim 4393   ` cfv 5255   cardccrd 7568   cfccf 7570
This theorem is referenced by:  cflim2  7889  cfss  7891  cfslb  7892
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-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-int 3863  df-iin 3908  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-lim 4397  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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