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Theorem cff 8128
Description: Cofinality is a function on the class of ordinal numbers to the class of cardinal numbers. (Contributed by Mario Carneiro, 15-Sep-2013.)
Assertion
Ref Expression
cff  |-  cf : On
--> On

Proof of Theorem cff
Dummy variables  x  y  z  w  v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-cf 7828 . 2  |-  cf  =  ( x  e.  On  |->  |^|
{ y  |  E. z ( y  =  ( card `  z
)  /\  ( z  C_  x  /\  A. w  e.  x  E. v  e.  z  w  C_  v
) ) } )
2 cardon 7831 . . . . . . 7  |-  ( card `  z )  e.  On
3 eleq1 2496 . . . . . . 7  |-  ( y  =  ( card `  z
)  ->  ( y  e.  On  <->  ( card `  z
)  e.  On ) )
42, 3mpbiri 225 . . . . . 6  |-  ( y  =  ( card `  z
)  ->  y  e.  On )
54adantr 452 . . . . 5  |-  ( ( y  =  ( card `  z )  /\  (
z  C_  x  /\  A. w  e.  x  E. v  e.  z  w  C_  v ) )  -> 
y  e.  On )
65exlimiv 1644 . . . 4  |-  ( E. z ( y  =  ( card `  z
)  /\  ( z  C_  x  /\  A. w  e.  x  E. v  e.  z  w  C_  v
) )  ->  y  e.  On )
76abssi 3418 . . 3  |-  { y  |  E. z ( y  =  ( card `  z )  /\  (
z  C_  x  /\  A. w  e.  x  E. v  e.  z  w  C_  v ) ) } 
C_  On
8 cflem 8126 . . . 4  |-  ( x  e.  On  ->  E. y E. z ( y  =  ( card `  z
)  /\  ( z  C_  x  /\  A. w  e.  x  E. v  e.  z  w  C_  v
) ) )
9 abn0 3646 . . . 4  |-  ( { y  |  E. z
( y  =  (
card `  z )  /\  ( z  C_  x  /\  A. w  e.  x  E. v  e.  z  w  C_  v ) ) }  =/=  (/)  <->  E. y E. z ( y  =  ( card `  z
)  /\  ( z  C_  x  /\  A. w  e.  x  E. v  e.  z  w  C_  v
) ) )
108, 9sylibr 204 . . 3  |-  ( x  e.  On  ->  { y  |  E. z ( y  =  ( card `  z )  /\  (
z  C_  x  /\  A. w  e.  x  E. v  e.  z  w  C_  v ) ) }  =/=  (/) )
11 oninton 4780 . . 3  |-  ( ( { y  |  E. z ( y  =  ( card `  z
)  /\  ( z  C_  x  /\  A. w  e.  x  E. v  e.  z  w  C_  v
) ) }  C_  On  /\  { y  |  E. z ( y  =  ( card `  z
)  /\  ( z  C_  x  /\  A. w  e.  x  E. v  e.  z  w  C_  v
) ) }  =/=  (/) )  ->  |^| { y  |  E. z ( y  =  ( card `  z )  /\  (
z  C_  x  /\  A. w  e.  x  E. v  e.  z  w  C_  v ) ) }  e.  On )
127, 10, 11sylancr 645 . 2  |-  ( x  e.  On  ->  |^| { y  |  E. z ( y  =  ( card `  z )  /\  (
z  C_  x  /\  A. w  e.  x  E. v  e.  z  w  C_  v ) ) }  e.  On )
131, 12fmpti 5892 1  |-  cf : On
--> On
Colors of variables: wff set class
Syntax hints:    /\ wa 359   E.wex 1550    = wceq 1652    e. wcel 1725   {cab 2422    =/= wne 2599   A.wral 2705   E.wrex 2706    C_ wss 3320   (/)c0 3628   |^|cint 4050   Oncon0 4581   -->wf 5450   ` cfv 5454   cardccrd 7822   cfccf 7824
This theorem is referenced by:  cfub  8129  cardcf  8132  cflecard  8133  cfle  8134  cflim2  8143  cfidm  8155
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-fv 5462  df-card 7826  df-cf 7828
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