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

Theorem cflim3 8134
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 4632 . . . 4  |-  ( Lim 
A  ->  Ord  A )
2 cflim3.1 . . . . 5  |-  A  e. 
_V
32elon 4582 . . . 4  |-  ( A  e.  On  <->  Ord  A )
41, 3sylibr 204 . . 3  |-  ( Lim 
A  ->  A  e.  On )
5 cfval 8119 . . 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 5734 . . . 4  |-  ( card `  x )  e.  _V
87dfiin2 4118 . . 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 2703 . . . . . 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 2876 . . . . . . . . . 10  |-  ( x  e.  { x  e. 
~P A  |  U. x  =  A }  <->  ( x  e.  ~P A  /\  U. x  =  A ) )
122elpw2 4356 . . . . . . . . . . . 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 8133 . . . . . . . . . . . 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 1636 . . . . . 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 2549 . . . 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 4047 . . 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 2480 . 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 2467 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 1550    = wceq 1652    e. wcel 1725   {cab 2421   A.wral 2697   E.wrex 2698   {crab 2701   _Vcvv 2948    C_ wss 3312   ~Pcpw 3791   U.cuni 4007   |^|cint 4042   |^|_ciin 4086   Ord word 4572   Oncon0 4573   Lim wlim 4574   ` cfv 5446   cardccrd 7814   cfccf 7816
This theorem is referenced by:  cflim2  8135  cfss  8137  cfslb  8138
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-pw 3793  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-int 4043  df-iin 4088  df-br 4205  df-opab 4259  df-mpt 4260  df-tr 4295  df-eprel 4486  df-id 4490  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577  df-lim 4578  df-xp 4876  df-rel 4877  df-cnv 4878  df-co 4879  df-dm 4880  df-iota 5410  df-fun 5448  df-fv 5454  df-cf 7820
  Copyright terms: Public domain W3C validator