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Theorem cardidm 7525
Description: The cardinality function is idempotent. Proposition 10.11 of [TakeutiZaring] p. 85. (Contributed by Mario Carneiro, 7-Jan-2013.)
Assertion
Ref Expression
cardidm  |-  ( card `  ( card `  A
) )  =  (
card `  A )

Proof of Theorem cardidm
StepHypRef Expression
1 cardid2 7519 . . . . . . . 8  |-  ( A  e.  dom  card  ->  (
card `  A )  ~~  A )
2 ensym 6843 . . . . . . . 8  |-  ( (
card `  A )  ~~  A  ->  A  ~~  ( card `  A )
)
31, 2syl 17 . . . . . . 7  |-  ( A  e.  dom  card  ->  A 
~~  ( card `  A
) )
4 entr 6846 . . . . . . . 8  |-  ( ( y  ~~  A  /\  A  ~~  ( card `  A
) )  ->  y  ~~  ( card `  A
) )
54expcom 426 . . . . . . 7  |-  ( A 
~~  ( card `  A
)  ->  ( y  ~~  A  ->  y  ~~  ( card `  A )
) )
63, 5syl 17 . . . . . 6  |-  ( A  e.  dom  card  ->  ( y  ~~  A  -> 
y  ~~  ( card `  A ) ) )
7 entr 6846 . . . . . . . 8  |-  ( ( y  ~~  ( card `  A )  /\  ( card `  A )  ~~  A )  ->  y  ~~  A )
87expcom 426 . . . . . . 7  |-  ( (
card `  A )  ~~  A  ->  ( y 
~~  ( card `  A
)  ->  y  ~~  A ) )
91, 8syl 17 . . . . . 6  |-  ( A  e.  dom  card  ->  ( y  ~~  ( card `  A )  ->  y  ~~  A ) )
106, 9impbid 185 . . . . 5  |-  ( A  e.  dom  card  ->  ( y  ~~  A  <->  y  ~~  ( card `  A )
) )
1110rabbidv 2732 . . . 4  |-  ( A  e.  dom  card  ->  { y  e.  On  | 
y  ~~  A }  =  { y  e.  On  |  y  ~~  ( card `  A ) } )
1211inteqd 3808 . . 3  |-  ( A  e.  dom  card  ->  |^|
{ y  e.  On  |  y  ~~  A }  =  |^| { y  e.  On  |  y  ~~  ( card `  A ) } )
13 cardval3 7518 . . 3  |-  ( A  e.  dom  card  ->  (
card `  A )  =  |^| { y  e.  On  |  y  ~~  A } )
14 cardon 7510 . . . 4  |-  ( card `  A )  e.  On
15 oncardval 7521 . . . 4  |-  ( (
card `  A )  e.  On  ->  ( card `  ( card `  A
) )  =  |^| { y  e.  On  | 
y  ~~  ( card `  A ) } )
1614, 15mp1i 13 . . 3  |-  ( A  e.  dom  card  ->  (
card `  ( card `  A ) )  = 
|^| { y  e.  On  |  y  ~~  ( card `  A ) } )
1712, 13, 163eqtr4rd 2299 . 2  |-  ( A  e.  dom  card  ->  (
card `  ( card `  A ) )  =  ( card `  A
) )
18 card0 7524 . . 3  |-  ( card `  (/) )  =  (/)
19 ndmfv 5451 . . . 4  |-  ( -.  A  e.  dom  card  -> 
( card `  A )  =  (/) )
2019fveq2d 5427 . . 3  |-  ( -.  A  e.  dom  card  -> 
( card `  ( card `  A ) )  =  ( card `  (/) ) )
2118, 20, 193eqtr4a 2314 . 2  |-  ( -.  A  e.  dom  card  -> 
( card `  ( card `  A ) )  =  ( card `  A
) )
2217, 21pm2.61i 158 1  |-  ( card `  ( card `  A
) )  =  (
card `  A )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    = wceq 1619    e. wcel 1621   {crab 2519   (/)c0 3397   |^|cint 3803   class class class wbr 3963   Oncon0 4329   dom cdm 4626   ` cfv 4638    ~~ cen 6793   cardccrd 7501
This theorem is referenced by:  oncard  7526  cardlim  7538  cardiun  7548  alephnbtwn2  7632  infenaleph  7651  dfac12k  7706  pwsdompw  7763  cardcf  7811  cfeq0  7815  cfflb  7818  alephval2  8127  cfpwsdom  8139  gch2  8234  tskcard  8336  hashcard  11280
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
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