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Theorem cfval2 8686
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 8673 . 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 5883 . . . 4  |-  ( card `  x )  e.  _V
32dfiin2 4328 . . 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 2779 . . . . . 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 3003 . . . . . . . . 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 selpw 3983 . . . . . . . . . 10  |-  ( x  e.  ~P A  <->  x  C_  A
)
76anbi1i 699 . . . . . . . . 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 ) )
85, 7bitri 252 . . . . . . . 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 ) )
98anbi2ci 700 . . . . . . 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 ) ) )
109exbii 1712 . . . . . 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 ) ) )
114, 10bitri 252 . . . . 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 ) ) )
1211abbii 2554 . . . 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 ) ) }
1312inteqi 4253 . . 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 ) ) }
143, 13eqtr2i 2450 . 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 )
151, 14syl6eq 2477 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 setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437   E.wex 1659    e. wcel 1867   {cab 2405   A.wral 2773   E.wrex 2774   {crab 2777    C_ wss 3433   ~Pcpw 3976   |^|cint 4249   |^|_ciin 4294   Oncon0 5434   ` cfv 5593   cardccrd 8366   cfccf 8368
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4540  ax-nul 4548  ax-pr 4653
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-ral 2778  df-rex 2779  df-rab 2782  df-v 3080  df-sbc 3297  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-int 4250  df-iin 4296  df-br 4418  df-opab 4477  df-mpt 4478  df-id 4761  df-xp 4852  df-rel 4853  df-cnv 4854  df-co 4855  df-dm 4856  df-iota 5557  df-fun 5595  df-fv 5601  df-cf 8372
This theorem is referenced by: (None)
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