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Theorem cardcf 7762
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 7757 . . . 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 2730 . . . . . . . . 9  |-  v  e. 
_V
3 eqeq1 2259 . . . . . . . . . . 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 2005 . . . . . . . . 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 2851 . . . . . . . 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 5377 . . . . . . . . . . . 12  |-  ( v  =  ( card `  y
)  ->  ( card `  v )  =  (
card `  ( card `  y ) ) )
8 cardidm 7476 . . . . . . . . . . . 12  |-  ( card `  ( card `  y
) )  =  (
card `  y )
97, 8syl6eq 2301 . . . . . . . . . . 11  |-  ( v  =  ( card `  y
)  ->  ( card `  v )  =  (
card `  y )
)
10 eqeq2 2262 . . . . . . . . . . 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 2023 . . . . . . . 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 7461 . . . . . . 7  |-  ( card `  v )  e.  On
1614, 15syl6eqelr 2342 . . . . . 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 3105 . . . . 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 5391 . . . . . . 7  |-  ( cf `  A )  e.  _V
191, 18syl6eqelr 2342 . . . . . 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 4065 . . . . . 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 4477 . . . . 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 2327 . . 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 5377 . . . . 5  |-  ( v  =  ( cf `  A
)  ->  ( card `  v )  =  (
card `  ( cf `  A ) ) )
26 id 21 . . . . 5  |-  ( v  =  ( cf `  A
)  ->  v  =  ( cf `  A ) )
2725, 26eqeq12d 2267 . . . 4  |-  ( v  =  ( cf `  A
)  ->  ( ( card `  v )  =  v  <->  ( card `  ( cf `  A ) )  =  ( cf `  A
) ) )
2827, 14vtoclga 2787 . . 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 7758 . . . . . 6  |-  cf : On
--> On
3130fdmi 5251 . . . . 5  |-  dom  cf  =  On
3231eleq2i 2317 . . . 4  |-  ( A  e.  dom  cf  <->  A  e.  On )
33 ndmfv 5405 . . . 4  |-  ( -.  A  e.  dom  cf  ->  ( cf `  A
)  =  (/) )
3432, 33sylnbir 300 . . 3  |-  ( -.  A  e.  On  ->  ( cf `  A )  =  (/) )
35 card0 7475 . . . 4  |-  ( card `  (/) )  =  (/)
36 fveq2 5377 . . . 4  |-  ( ( cf `  A )  =  (/)  ->  ( card `  ( cf `  A
) )  =  (
card `  (/) ) )
37 id 21 . . . 4  |-  ( ( cf `  A )  =  (/)  ->  ( cf `  A )  =  (/) )
3835, 36, 373eqtr4a 2311 . . 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 2239    =/= wne 2412   A.wral 2509   E.wrex 2510   _Vcvv 2727    C_ wss 3078   (/)c0 3362   |^|cint 3760   Oncon0 4285   dom cdm 4580   ` cfv 4592   cardccrd 7452   cfccf 7454
This theorem is referenced by:  cfon  7765  winacard  8194
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 1926  ax-ext 2234  ax-sep 4038  ax-nul 4046  ax-pow 4082  ax-pr 4108  ax-un 4403
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 1883  df-eu 2118  df-mo 2119  df-clab 2240  df-cleq 2246  df-clel 2249  df-nfc 2374  df-ne 2414  df-ral 2513  df-rex 2514  df-rab 2516  df-v 2729  df-sbc 2922  df-dif 3081  df-un 3083  df-in 3085  df-ss 3089  df-pss 3091  df-nul 3363  df-if 3471  df-pw 3532  df-sn 3550  df-pr 3551  df-tp 3552  df-op 3553  df-uni 3728  df-int 3761  df-br 3921  df-opab 3975  df-mpt 3976  df-tr 4011  df-eprel 4198  df-id 4202  df-po 4207  df-so 4208  df-fr 4245  df-we 4247  df-ord 4288  df-on 4289  df-xp 4594  df-rel 4595  df-cnv 4596  df-co 4597  df-dm 4598  df-rn 4599  df-res 4600  df-ima 4601  df-fun 4602  df-fn 4603  df-f 4604  df-f1 4605  df-fo 4606  df-f1o 4607  df-fv 4608  df-er 6546  df-en 6750  df-card 7456  df-cf 7458
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