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Theorem cardcf 7811
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
StepHypRef Expression
1 cfval 7806 . . . 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 2743 . . . . . . . . 9  |-  v  e. 
_V
3 eqeq1 2262 . . . . . . . . . . 11  |-  ( x  =  v  ->  (
x  =  ( card `  y )  <->  v  =  ( card `  y )
) )
43anbi1d 688 . . . . . . . . . 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 2006 . . . . . . . . 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 2865 . . . . . . . 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 5423 . . . . . . . . . . . 12  |-  ( v  =  ( card `  y
)  ->  ( card `  v )  =  (
card `  ( card `  y ) ) )
8 cardidm 7525 . . . . . . . . . . . 12  |-  ( card `  ( card `  y
) )  =  (
card `  y )
97, 8syl6eq 2304 . . . . . . . . . . 11  |-  ( v  =  ( card `  y
)  ->  ( card `  v )  =  (
card `  y )
)
10 eqeq2 2265 . . . . . . . . . . 11  |-  ( v  =  ( card `  y
)  ->  ( ( card `  v )  =  v  <->  ( card `  v
)  =  ( card `  y ) ) )
119, 10mpbird 225 . . . . . . . . . 10  |-  ( v  =  ( card `  y
)  ->  ( card `  v )  =  v )
1211adantr 453 . . . . . . . . 9  |-  ( ( v  =  ( card `  y )  /\  (
y  C_  A  /\  A. z  e.  A  E. w  e.  y  z  C_  w ) )  -> 
( card `  v )  =  v )
1312exlimiv 2024 . . . . . . . 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 189 . . . . . . 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 7510 . . . . . . 7  |-  ( card `  v )  e.  On
1614, 15syl6eqelr 2345 . . . . . 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 3126 . . . . 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 5437 . . . . . . 7  |-  ( cf `  A )  e.  _V
191, 18syl6eqelr 2345 . . . . . 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 4109 . . . . . 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 205 . . . . 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 4523 . . . . 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 647 . . . 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 2330 . . 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 5423 . . . . 5  |-  ( v  =  ( cf `  A
)  ->  ( card `  v )  =  (
card `  ( cf `  A ) ) )
26 id 21 . . . . 5  |-  ( v  =  ( cf `  A
)  ->  v  =  ( cf `  A ) )
2725, 26eqeq12d 2270 . . . 4  |-  ( v  =  ( cf `  A
)  ->  ( ( card `  v )  =  v  <->  ( card `  ( cf `  A ) )  =  ( cf `  A
) ) )
2827, 14vtoclga 2800 . . 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 17 . 2  |-  ( A  e.  On  ->  ( card `  ( cf `  A
) )  =  ( cf `  A ) )
30 cff 7807 . . . . . 6  |-  cf : On
--> On
3130fdmi 5297 . . . . 5  |-  dom  cf  =  On
3231eleq2i 2320 . . . 4  |-  ( A  e.  dom  cf  <->  A  e.  On )
33 ndmfv 5451 . . . 4  |-  ( -.  A  e.  dom  cf  ->  ( cf `  A
)  =  (/) )
3432, 33sylnbir 300 . . 3  |-  ( -.  A  e.  On  ->  ( cf `  A )  =  (/) )
35 card0 7524 . . . 4  |-  ( card `  (/) )  =  (/)
36 fveq2 5423 . . . 4  |-  ( ( cf `  A )  =  (/)  ->  ( card `  ( cf `  A
) )  =  (
card `  (/) ) )
37 id 21 . . . 4  |-  ( ( cf `  A )  =  (/)  ->  ( cf `  A )  =  (/) )
3835, 36, 373eqtr4a 2314 . . 3  |-  ( ( cf `  A )  =  (/)  ->  ( card `  ( cf `  A
) )  =  ( cf `  A ) )
3934, 38syl 17 . 2  |-  ( -.  A  e.  On  ->  (
card `  ( cf `  A ) )  =  ( cf `  A
) )
4029, 39pm2.61i 158 1  |-  ( card `  ( cf `  A
) )  =  ( cf `  A )
Colors of variables: wff set class
Syntax hints:   -. wn 5    /\ wa 360   E.wex 1537    = wceq 1619    e. wcel 1621   {cab 2242    =/= wne 2419   A.wral 2516   E.wrex 2517   _Vcvv 2740    C_ wss 3094   (/)c0 3397   |^|cint 3803   Oncon0 4329   dom cdm 4626   ` cfv 4638   cardccrd 7501   cfccf 7503
This theorem is referenced by:  cfon  7814  winacard  8247
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-13 1625  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2237  ax-sep 4081  ax-nul 4089  ax-pow 4126  ax-pr 4152  ax-un 4449
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3or 940  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2121  df-mo 2122  df-clab 2243  df-cleq 2249  df-clel 2252  df-nfc 2381  df-ne 2421  df-ral 2520  df-rex 2521  df-rab 2523  df-v 2742  df-sbc 2936  df-dif 3097  df-un 3099  df-in 3101  df-ss 3108  df-pss 3110  df-nul 3398  df-if 3507  df-pw 3568  df-sn 3587  df-pr 3588  df-tp 3589  df-op 3590  df-uni 3769  df-int 3804  df-br 3964  df-opab 4018  df-mpt 4019  df-tr 4054  df-eprel 4242  df-id 4246  df-po 4251  df-so 4252  df-fr 4289  df-we 4291  df-ord 4332  df-on 4333  df-xp 4640  df-rel 4641  df-cnv 4642  df-co 4643  df-dm 4644  df-rn 4645  df-res 4646  df-ima 4647  df-fun 4648  df-fn 4649  df-f 4650  df-f1 4651  df-fo 4652  df-f1o 4653  df-fv 4654  df-er 6593  df-en 6797  df-card 7505  df-cf 7507
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