MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cfval2 Structured version   Unicode version

Theorem cfval2 8142
Description: Another expression for the cofinality function. (Contributed by Mario Carneiro, 28-Feb-2013.)
Assertion
Ref Expression
cfval2  |-  ( A  e.  On  ->  ( cf `  A )  = 
|^|_ x  e.  { x  e.  ~P A  |  A. z  e.  A  E. w  e.  x  z  C_  w }  ( card `  x ) )
Distinct variable group:    w, A, x, z

Proof of Theorem cfval2
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 cfval 8129 . 2  |-  ( 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 ) ) } )
2 fvex 5744 . . . 4  |-  ( card `  x )  e.  _V
32dfiin2 4128 . . 3  |-  |^|_ x  e.  { x  e.  ~P A  |  A. z  e.  A  E. w  e.  x  z  C_  w }  ( card `  x )  =  |^| { y  |  E. x  e.  { x  e.  ~P A  |  A. z  e.  A  E. w  e.  x  z  C_  w } y  =  (
card `  x ) }
4 df-rex 2713 . . . . . 6  |-  ( E. x  e.  { x  e.  ~P A  |  A. z  e.  A  E. w  e.  x  z  C_  w } y  =  ( card `  x
)  <->  E. x ( x  e.  { x  e. 
~P A  |  A. z  e.  A  E. w  e.  x  z  C_  w }  /\  y  =  ( card `  x
) ) )
5 rabid 2886 . . . . . . . . 9  |-  ( x  e.  { x  e. 
~P A  |  A. z  e.  A  E. w  e.  x  z  C_  w }  <->  ( x  e.  ~P A  /\  A. z  e.  A  E. w  e.  x  z  C_  w ) )
6 vex 2961 . . . . . . . . . . 11  |-  x  e. 
_V
76elpw 3807 . . . . . . . . . 10  |-  ( x  e.  ~P A  <->  x  C_  A
)
87anbi1i 678 . . . . . . . . 9  |-  ( ( x  e.  ~P A  /\  A. z  e.  A  E. w  e.  x  z  C_  w )  <->  ( x  C_  A  /\  A. z  e.  A  E. w  e.  x  z  C_  w ) )
95, 8bitri 242 . . . . . . . 8  |-  ( x  e.  { x  e. 
~P A  |  A. z  e.  A  E. w  e.  x  z  C_  w }  <->  ( x  C_  A  /\  A. z  e.  A  E. w  e.  x  z  C_  w ) )
109anbi2ci 679 . . . . . . 7  |-  ( ( x  e.  { x  e.  ~P A  |  A. z  e.  A  E. w  e.  x  z  C_  w }  /\  y  =  ( card `  x
) )  <->  ( y  =  ( card `  x
)  /\  ( x  C_  A  /\  A. z  e.  A  E. w  e.  x  z  C_  w ) ) )
1110exbii 1593 . . . . . 6  |-  ( E. x ( x  e. 
{ x  e.  ~P A  |  A. z  e.  A  E. w  e.  x  z  C_  w }  /\  y  =  ( card `  x
) )  <->  E. x
( y  =  (
card `  x )  /\  ( x  C_  A  /\  A. z  e.  A  E. w  e.  x  z  C_  w ) ) )
124, 11bitri 242 . . . . 5  |-  ( E. x  e.  { x  e.  ~P A  |  A. z  e.  A  E. w  e.  x  z  C_  w } y  =  ( card `  x
)  <->  E. x ( y  =  ( card `  x
)  /\  ( x  C_  A  /\  A. z  e.  A  E. w  e.  x  z  C_  w ) ) )
1312abbii 2550 . . . 4  |-  { y  |  E. x  e. 
{ x  e.  ~P A  |  A. z  e.  A  E. w  e.  x  z  C_  w } y  =  (
card `  x ) }  =  { y  |  E. x ( y  =  ( card `  x
)  /\  ( x  C_  A  /\  A. z  e.  A  E. w  e.  x  z  C_  w ) ) }
1413inteqi 4056 . . 3  |-  |^| { y  |  E. x  e. 
{ x  e.  ~P A  |  A. z  e.  A  E. w  e.  x  z  C_  w } y  =  (
card `  x ) }  =  |^| { y  |  E. x ( y  =  ( card `  x )  /\  (
x  C_  A  /\  A. z  e.  A  E. w  e.  x  z  C_  w ) ) }
153, 14eqtr2i 2459 . 2  |-  |^| { 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  |  A. z  e.  A  E. w  e.  x  z  C_  w }  ( card `  x )
161, 15syl6eq 2486 1  |-  ( A  e.  On  ->  ( cf `  A )  = 
|^|_ x  e.  { x  e.  ~P A  |  A. z  e.  A  E. w  e.  x  z  C_  w }  ( card `  x ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360   E.wex 1551    = wceq 1653    e. wcel 1726   {cab 2424   A.wral 2707   E.wrex 2708   {crab 2711    C_ wss 3322   ~Pcpw 3801   |^|cint 4052   |^|_ciin 4096   Oncon0 4583   ` cfv 5456   cardccrd 7824   cfccf 7826
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4332  ax-nul 4340  ax-pr 4405
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-int 4053  df-iin 4098  df-br 4215  df-opab 4269  df-mpt 4270  df-id 4500  df-xp 4886  df-rel 4887  df-cnv 4888  df-co 4889  df-dm 4890  df-iota 5420  df-fun 5458  df-fv 5464  df-cf 7830
  Copyright terms: Public domain W3C validator