MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cflim3 Unicode version

Theorem cflim3 8075
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 4581 . . . 4  |-  ( Lim 
A  ->  Ord  A )
2 cflim3.1 . . . . 5  |-  A  e. 
_V
32elon 4531 . . . 4  |-  ( A  e.  On  <->  Ord  A )
41, 3sylibr 204 . . 3  |-  ( Lim 
A  ->  A  e.  On )
5 cfval 8060 . . 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 16 . 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 5682 . . . 4  |-  ( card `  x )  e.  _V
87dfiin2 4068 . . 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 2655 . . . . . 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 438 . . . . . . . 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 2827 . . . . . . . . . 10  |-  ( x  e.  { x  e. 
~P A  |  U. x  =  A }  <->  ( x  e.  ~P A  /\  U. x  =  A ) )
122elpw2 4305 . . . . . . . . . . . 12  |-  ( x  e.  ~P A  <->  x  C_  A
)
1312anbi1i 677 . . . . . . . . . . 11  |-  ( ( x  e.  ~P A  /\  U. x  =  A )  <->  ( x  C_  A  /\  U. x  =  A ) )
14 coflim 8074 . . . . . . . . . . . 12  |-  ( ( Lim  A  /\  x  C_  A )  ->  ( U. x  =  A  <->  A. z  e.  A  E. w  e.  x  z  C_  w ) )
1514pm5.32da 623 . . . . . . . . . . 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 249 . . . . . . . . . 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 249 . . . . . . . . 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 685 . . . . . . . 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 249 . . . . . . 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 1633 . . . . . 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 249 . . . . 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 2501 . . . 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 3997 . . 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 2432 . 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 2419 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 359   E.wex 1547    = wceq 1649    e. wcel 1717   {cab 2373   A.wral 2649   E.wrex 2650   {crab 2653   _Vcvv 2899    C_ wss 3263   ~Pcpw 3742   U.cuni 3957   |^|cint 3992   |^|_ciin 4036   Ord word 4521   Oncon0 4522   Lim wlim 4523   ` cfv 5394   cardccrd 7755   cfccf 7757
This theorem is referenced by:  cflim2  8076  cfss  8078  cfslb  8079
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 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2368  ax-sep 4271  ax-nul 4279  ax-pr 4344
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 2242  df-mo 2243  df-clab 2374  df-cleq 2380  df-clel 2383  df-nfc 2512  df-ne 2552  df-ral 2654  df-rex 2655  df-rab 2658  df-v 2901  df-sbc 3105  df-dif 3266  df-un 3268  df-in 3270  df-ss 3277  df-nul 3572  df-if 3683  df-pw 3744  df-sn 3763  df-pr 3764  df-op 3766  df-uni 3958  df-int 3993  df-iin 4038  df-br 4154  df-opab 4208  df-mpt 4209  df-tr 4244  df-eprel 4435  df-id 4439  df-po 4444  df-so 4445  df-fr 4482  df-we 4484  df-ord 4525  df-on 4526  df-lim 4527  df-xp 4824  df-rel 4825  df-cnv 4826  df-co 4827  df-dm 4828  df-iota 5358  df-fun 5396  df-fv 5402  df-cf 7761
  Copyright terms: Public domain W3C validator