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Theorem cff 7874
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 7574 . 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 7577 . . . . . . 7  |-  ( card `  z )  e.  On
3 eleq1 2343 . . . . . . 7  |-  ( y  =  ( card `  z
)  ->  ( y  e.  On  <->  ( card `  z
)  e.  On ) )
42, 3mpbiri 224 . . . . . 6  |-  ( y  =  ( card `  z
)  ->  y  e.  On )
54adantr 451 . . . . 5  |-  ( ( y  =  ( card `  z )  /\  (
z  C_  x  /\  A. w  e.  x  E. v  e.  z  w  C_  v ) )  -> 
y  e.  On )
65exlimiv 1666 . . . 4  |-  ( E. z ( y  =  ( card `  z
)  /\  ( z  C_  x  /\  A. w  e.  x  E. v  e.  z  w  C_  v
) )  ->  y  e.  On )
76abssi 3248 . . 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 7872 . . . 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 3473 . . . 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 203 . . 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 4591 . . 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 644 . 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 5683 1  |-  cf : On
--> On
Colors of variables: wff set class
Syntax hints:    /\ wa 358   E.wex 1528    = wceq 1623    e. wcel 1684   {cab 2269    =/= wne 2446   A.wral 2543   E.wrex 2544    C_ wss 3152   (/)c0 3455   |^|cint 3862   Oncon0 4392   -->wf 5251   ` cfv 5255   cardccrd 7568   cfccf 7570
This theorem is referenced by:  cfub  7875  cardcf  7878  cflecard  7879  cfle  7880  cflim2  7889  cfidm  7901
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-card 7572  df-cf 7574
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