MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  cardcf Unicode version

Theorem cardcf 7894
Description: Cofinality is a cardinal number. Proposition 11.11 of [TakeutiZaring] p. 103. (Contributed by NM, 24-Apr-2004.) (Revised by Mario Carneiro, 15-Sep-2013.)
Assertion
Ref Expression
cardcf  |-  ( card `  ( cf `  A
) )  =  ( cf `  A )

Proof of Theorem cardcf
Dummy variables  x  y  z  w  v are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cfval 7889 . . . 4  |-  ( A  e.  On  ->  ( cf `  A )  = 
|^| { x  |  E. y ( x  =  ( card `  y
)  /\  ( y  C_  A  /\  A. z  e.  A  E. w  e.  y  z  C_  w ) ) } )
2 vex 2804 . . . . . . . . 9  |-  v  e. 
_V
3 eqeq1 2302 . . . . . . . . . . 11  |-  ( x  =  v  ->  (
x  =  ( card `  y )  <->  v  =  ( card `  y )
) )
43anbi1d 685 . . . . . . . . . 10  |-  ( x  =  v  ->  (
( x  =  (
card `  y )  /\  ( y  C_  A  /\  A. z  e.  A  E. w  e.  y 
z  C_  w )
)  <->  ( v  =  ( card `  y
)  /\  ( y  C_  A  /\  A. z  e.  A  E. w  e.  y  z  C_  w ) ) ) )
54exbidv 1616 . . . . . . . . 9  |-  ( x  =  v  ->  ( E. y ( x  =  ( card `  y
)  /\  ( y  C_  A  /\  A. z  e.  A  E. w  e.  y  z  C_  w ) )  <->  E. y
( v  =  (
card `  y )  /\  ( y  C_  A  /\  A. z  e.  A  E. w  e.  y 
z  C_  w )
) ) )
62, 5elab 2927 . . . . . . . 8  |-  ( v  e.  { x  |  E. y ( x  =  ( card `  y
)  /\  ( y  C_  A  /\  A. z  e.  A  E. w  e.  y  z  C_  w ) ) }  <->  E. y ( v  =  ( card `  y
)  /\  ( y  C_  A  /\  A. z  e.  A  E. w  e.  y  z  C_  w ) ) )
7 fveq2 5541 . . . . . . . . . . . 12  |-  ( v  =  ( card `  y
)  ->  ( card `  v )  =  (
card `  ( card `  y ) ) )
8 cardidm 7608 . . . . . . . . . . . 12  |-  ( card `  ( card `  y
) )  =  (
card `  y )
97, 8syl6eq 2344 . . . . . . . . . . 11  |-  ( v  =  ( card `  y
)  ->  ( card `  v )  =  (
card `  y )
)
10 eqeq2 2305 . . . . . . . . . . 11  |-  ( v  =  ( card `  y
)  ->  ( ( card `  v )  =  v  <->  ( card `  v
)  =  ( card `  y ) ) )
119, 10mpbird 223 . . . . . . . . . 10  |-  ( v  =  ( card `  y
)  ->  ( card `  v )  =  v )
1211adantr 451 . . . . . . . . 9  |-  ( ( v  =  ( card `  y )  /\  (
y  C_  A  /\  A. z  e.  A  E. w  e.  y  z  C_  w ) )  -> 
( card `  v )  =  v )
1312exlimiv 1624 . . . . . . . 8  |-  ( E. y ( v  =  ( card `  y
)  /\  ( y  C_  A  /\  A. z  e.  A  E. w  e.  y  z  C_  w ) )  -> 
( card `  v )  =  v )
146, 13sylbi 187 . . . . . . 7  |-  ( v  e.  { x  |  E. y ( x  =  ( card `  y
)  /\  ( y  C_  A  /\  A. z  e.  A  E. w  e.  y  z  C_  w ) ) }  ->  ( card `  v
)  =  v )
15 cardon 7593 . . . . . . 7  |-  ( card `  v )  e.  On
1614, 15syl6eqelr 2385 . . . . . 6  |-  ( v  e.  { x  |  E. y ( x  =  ( card `  y
)  /\  ( y  C_  A  /\  A. z  e.  A  E. w  e.  y  z  C_  w ) ) }  ->  v  e.  On )
1716ssriv 3197 . . . . 5  |-  { x  |  E. y ( x  =  ( card `  y
)  /\  ( y  C_  A  /\  A. z  e.  A  E. w  e.  y  z  C_  w ) ) } 
C_  On
18 fvex 5555 . . . . . . 7  |-  ( cf `  A )  e.  _V
191, 18syl6eqelr 2385 . . . . . 6  |-  ( A  e.  On  ->  |^| { x  |  E. y ( x  =  ( card `  y
)  /\  ( y  C_  A  /\  A. z  e.  A  E. w  e.  y  z  C_  w ) ) }  e.  _V )
20 intex 4183 . . . . . 6  |-  ( { x  |  E. y
( x  =  (
card `  y )  /\  ( y  C_  A  /\  A. z  e.  A  E. w  e.  y 
z  C_  w )
) }  =/=  (/)  <->  |^| { x  |  E. y ( x  =  ( card `  y
)  /\  ( y  C_  A  /\  A. z  e.  A  E. w  e.  y  z  C_  w ) ) }  e.  _V )
2119, 20sylibr 203 . . . . 5  |-  ( A  e.  On  ->  { x  |  E. y ( x  =  ( card `  y
)  /\  ( y  C_  A  /\  A. z  e.  A  E. w  e.  y  z  C_  w ) ) }  =/=  (/) )
22 onint 4602 . . . . 5  |-  ( ( { x  |  E. y ( x  =  ( card `  y
)  /\  ( y  C_  A  /\  A. z  e.  A  E. w  e.  y  z  C_  w ) ) } 
C_  On  /\  { x  |  E. y ( x  =  ( card `  y
)  /\  ( y  C_  A  /\  A. z  e.  A  E. w  e.  y  z  C_  w ) ) }  =/=  (/) )  ->  |^| { x  |  E. y ( x  =  ( card `  y
)  /\  ( y  C_  A  /\  A. z  e.  A  E. w  e.  y  z  C_  w ) ) }  e.  { x  |  E. y ( x  =  ( card `  y
)  /\  ( y  C_  A  /\  A. z  e.  A  E. w  e.  y  z  C_  w ) ) } )
2317, 21, 22sylancr 644 . . . 4  |-  ( A  e.  On  ->  |^| { x  |  E. y ( x  =  ( card `  y
)  /\  ( y  C_  A  /\  A. z  e.  A  E. w  e.  y  z  C_  w ) ) }  e.  { x  |  E. y ( x  =  ( card `  y
)  /\  ( y  C_  A  /\  A. z  e.  A  E. w  e.  y  z  C_  w ) ) } )
241, 23eqeltrd 2370 . . 3  |-  ( A  e.  On  ->  ( cf `  A )  e. 
{ x  |  E. y ( x  =  ( card `  y
)  /\  ( y  C_  A  /\  A. z  e.  A  E. w  e.  y  z  C_  w ) ) } )
25 fveq2 5541 . . . . 5  |-  ( v  =  ( cf `  A
)  ->  ( card `  v )  =  (
card `  ( cf `  A ) ) )
26 id 19 . . . . 5  |-  ( v  =  ( cf `  A
)  ->  v  =  ( cf `  A ) )
2725, 26eqeq12d 2310 . . . 4  |-  ( v  =  ( cf `  A
)  ->  ( ( card `  v )  =  v  <->  ( card `  ( cf `  A ) )  =  ( cf `  A
) ) )
2827, 14vtoclga 2862 . . 3  |-  ( ( cf `  A )  e.  { x  |  E. y ( x  =  ( card `  y
)  /\  ( y  C_  A  /\  A. z  e.  A  E. w  e.  y  z  C_  w ) ) }  ->  ( card `  ( cf `  A ) )  =  ( cf `  A
) )
2924, 28syl 15 . 2  |-  ( A  e.  On  ->  ( card `  ( cf `  A
) )  =  ( cf `  A ) )
30 cff 7890 . . . . . 6  |-  cf : On
--> On
3130fdmi 5410 . . . . 5  |-  dom  cf  =  On
3231eleq2i 2360 . . . 4  |-  ( A  e.  dom  cf  <->  A  e.  On )
33 ndmfv 5568 . . . 4  |-  ( -.  A  e.  dom  cf  ->  ( cf `  A
)  =  (/) )
3432, 33sylnbir 298 . . 3  |-  ( -.  A  e.  On  ->  ( cf `  A )  =  (/) )
35 card0 7607 . . . 4  |-  ( card `  (/) )  =  (/)
36 fveq2 5541 . . . 4  |-  ( ( cf `  A )  =  (/)  ->  ( card `  ( cf `  A
) )  =  (
card `  (/) ) )
37 id 19 . . . 4  |-  ( ( cf `  A )  =  (/)  ->  ( cf `  A )  =  (/) )
3835, 36, 373eqtr4a 2354 . . 3  |-  ( ( cf `  A )  =  (/)  ->  ( card `  ( cf `  A
) )  =  ( cf `  A ) )
3934, 38syl 15 . 2  |-  ( -.  A  e.  On  ->  (
card `  ( cf `  A ) )  =  ( cf `  A
) )
4029, 39pm2.61i 156 1  |-  ( card `  ( cf `  A
) )  =  ( cf `  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 358   E.wex 1531    = wceq 1632    e. wcel 1696   {cab 2282    =/= wne 2459   A.wral 2556   E.wrex 2557   _Vcvv 2801    C_ wss 3165   (/)c0 3468   |^|cint 3878   Oncon0 4408   dom cdm 4705   ` cfv 5271   cardccrd 7584   cfccf 7586
This theorem is referenced by:  cfon  7897  winacard  8330
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-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-er 6676  df-en 6880  df-card 7588  df-cf 7590
  Copyright terms: Public domain W3C validator