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

Theorem cfval2 7902
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 7889 . 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 5555 . . . 4  |-  ( card `  x )  e.  _V
32dfiin2 3954 . . 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 2562 . . . . . 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 2729 . . . . . . . . 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 2804 . . . . . . . . . . 11  |-  x  e. 
_V
76elpw 3644 . . . . . . . . . 10  |-  ( x  e.  ~P A  <->  x  C_  A
)
87anbi1i 676 . . . . . . . . 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 240 . . . . . . . 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 677 . . . . . . 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 1572 . . . . . 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 240 . . . . 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 2408 . . . 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 3882 . . 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 2317 . 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 2344 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 358   E.wex 1531    = wceq 1632    e. wcel 1696   {cab 2282   A.wral 2556   E.wrex 2557   {crab 2560    C_ wss 3165   ~Pcpw 3638   |^|cint 3878   |^|_ciin 3922   Oncon0 4408   ` cfv 5271   cardccrd 7584   cfccf 7586
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-id 4325  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
  Copyright terms: Public domain W3C validator