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Theorem cardcf 7874
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 7869 . . . 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 2792 . . . . . . . . 9  |-  v  e. 
_V
3 eqeq1 2290 . . . . . . . . . . 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 1612 . . . . . . . . 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 2915 . . . . . . . 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 5486 . . . . . . . . . . . 12  |-  ( v  =  ( card `  y
)  ->  ( card `  v )  =  (
card `  ( card `  y ) ) )
8 cardidm 7588 . . . . . . . . . . . 12  |-  ( card `  ( card `  y
) )  =  (
card `  y )
97, 8syl6eq 2332 . . . . . . . . . . 11  |-  ( v  =  ( card `  y
)  ->  ( card `  v )  =  (
card `  y )
)
10 eqeq2 2293 . . . . . . . . . . 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 1667 . . . . . . . 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 7573 . . . . . . 7  |-  ( card `  v )  e.  On
1614, 15syl6eqelr 2373 . . . . . 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 3185 . . . . 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 5500 . . . . . . 7  |-  ( cf `  A )  e.  _V
191, 18syl6eqelr 2373 . . . . . 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 4170 . . . . . 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 4585 . . . . 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 2358 . . 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 5486 . . . . 5  |-  ( v  =  ( cf `  A
)  ->  ( card `  v )  =  (
card `  ( cf `  A ) ) )
26 id 19 . . . . 5  |-  ( v  =  ( cf `  A
)  ->  v  =  ( cf `  A ) )
2725, 26eqeq12d 2298 . . . 4  |-  ( v  =  ( cf `  A
)  ->  ( ( card `  v )  =  v  <->  ( card `  ( cf `  A ) )  =  ( cf `  A
) ) )
2827, 14vtoclga 2850 . . 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 7870 . . . . . 6  |-  cf : On
--> On
3130fdmi 5360 . . . . 5  |-  dom  cf  =  On
3231eleq2i 2348 . . . 4  |-  ( A  e.  dom  cf  <->  A  e.  On )
33 ndmfv 5514 . . . 4  |-  ( -.  A  e.  dom  cf  ->  ( cf `  A
)  =  (/) )
3432, 33sylnbir 298 . . 3  |-  ( -.  A  e.  On  ->  ( cf `  A )  =  (/) )
35 card0 7587 . . . 4  |-  ( card `  (/) )  =  (/)
36 fveq2 5486 . . . 4  |-  ( ( cf `  A )  =  (/)  ->  ( card `  ( cf `  A
) )  =  (
card `  (/) ) )
37 id 19 . . . 4  |-  ( ( cf `  A )  =  (/)  ->  ( cf `  A )  =  (/) )
3835, 36, 373eqtr4a 2342 . . 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 1528    = wceq 1623    e. wcel 1685   {cab 2270    =/= wne 2447   A.wral 2544   E.wrex 2545   _Vcvv 2789    C_ wss 3153   (/)c0 3456   |^|cint 3863   Oncon0 4391    dom cdm 4688   ` cfv 5221   cardccrd 7564   cfccf 7566
This theorem is referenced by:  cfon  7877  winacard  8310
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 1636  ax-8 1644  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pow 4187  ax-pr 4213  ax-un 4511
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 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-rab 2553  df-v 2791  df-sbc 2993  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pss 3169  df-nul 3457  df-if 3567  df-pw 3628  df-sn 3647  df-pr 3648  df-tp 3649  df-op 3650  df-uni 3829  df-int 3864  df-br 4025  df-opab 4079  df-mpt 4080  df-tr 4115  df-eprel 4304  df-id 4308  df-po 4313  df-so 4314  df-fr 4351  df-we 4353  df-ord 4394  df-on 4395  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-er 6656  df-en 6860  df-card 7568  df-cf 7570
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