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Theorem cfval2 8708
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 8695 . 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 5889 . . . 4  |-  ( card `  x )  e.  _V
32dfiin2 4304 . . 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 2762 . . . . . 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 2953 . . . . . . . . 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 3949 . . . . . . . . . 10  |-  ( x  e.  ~P A  <->  x  C_  A
)
76anbi1i 709 . . . . . . . . 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 257 . . . . . . . 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 710 . . . . . . 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 1726 . . . . . 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 257 . . . . 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 2587 . . . 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 4230 . . 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 2494 . 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 2521 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 376    = wceq 1452   E.wex 1671    e. wcel 1904   {cab 2457   A.wral 2756   E.wrex 2757   {crab 2760    C_ wss 3390   ~Pcpw 3942   |^|cint 4226   |^|_ciin 4270   Oncon0 5430   ` cfv 5589   cardccrd 8387   cfccf 8389
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-int 4227  df-iin 4272  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-iota 5553  df-fun 5591  df-fv 5597  df-cf 8393
This theorem is referenced by: (None)
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