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Theorem cff 7890
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 7590 . 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 7593 . . . . . . 7  |-  ( card `  z )  e.  On
3 eleq1 2356 . . . . . . 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 1624 . . . 4  |-  ( E. z ( y  =  ( card `  z
)  /\  ( z  C_  x  /\  A. w  e.  x  E. v  e.  z  w  C_  v
) )  ->  y  e.  On )
76abssi 3261 . . 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 7888 . . . 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 3486 . . . 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 4607 . . 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 5699 1  |-  cf : On
--> On
Colors of variables: wff set class
Syntax hints:    /\ wa 358   E.wex 1531    = wceq 1632    e. wcel 1696   {cab 2282    =/= wne 2459   A.wral 2556   E.wrex 2557    C_ wss 3165   (/)c0 3468   |^|cint 3878   Oncon0 4408   -->wf 5267   ` cfv 5271   cardccrd 7584   cfccf 7586
This theorem is referenced by:  cfub  7891  cardcf  7894  cflecard  7895  cfle  7896  cflim2  7905  cfidm  7917
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pow 4204  ax-pr 4230  ax-un 4528
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 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-fv 5279  df-card 7588  df-cf 7590
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