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Theorem cfval2 8096
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 8083 . 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 5701 . . . 4  |-  ( card `  x )  e.  _V
32dfiin2 4086 . . 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 2672 . . . . . 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 2844 . . . . . . . . 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 2919 . . . . . . . . . . 11  |-  x  e. 
_V
76elpw 3765 . . . . . . . . . 10  |-  ( x  e.  ~P A  <->  x  C_  A
)
87anbi1i 677 . . . . . . . . 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 241 . . . . . . . 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 678 . . . . . . 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 1589 . . . . . 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 241 . . . . 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 2516 . . . 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 4014 . . 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 2425 . 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 2452 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 359   E.wex 1547    = wceq 1649    e. wcel 1721   {cab 2390   A.wral 2666   E.wrex 2667   {crab 2670    C_ wss 3280   ~Pcpw 3759   |^|cint 4010   |^|_ciin 4054   Oncon0 4541   ` cfv 5413   cardccrd 7778   cfccf 7780
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 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363
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 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-pw 3761  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-int 4011  df-iin 4056  df-br 4173  df-opab 4227  df-mpt 4228  df-id 4458  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-iota 5377  df-fun 5415  df-fv 5421  df-cf 7784
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