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Theorem cardidm 7546
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 7540 . . . . . . . 8  |-  ( A  e.  dom  card  ->  (
card `  A )  ~~  A )
2 ensym 6864 . . . . . . . 8  |-  ( (
card `  A )  ~~  A  ->  A  ~~  ( card `  A )
)
31, 2syl 17 . . . . . . 7  |-  ( A  e.  dom  card  ->  A 
~~  ( card `  A
) )
4 entr 6867 . . . . . . . 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 6867 . . . . . . . 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 2749 . . . 4  |-  ( A  e.  dom  card  ->  { y  e.  On  | 
y  ~~  A }  =  { y  e.  On  |  y  ~~  ( card `  A ) } )
1211inteqd 3827 . . 3  |-  ( A  e.  dom  card  ->  |^|
{ y  e.  On  |  y  ~~  A }  =  |^| { y  e.  On  |  y  ~~  ( card `  A ) } )
13 cardval3 7539 . . 3  |-  ( A  e.  dom  card  ->  (
card `  A )  =  |^| { y  e.  On  |  y  ~~  A } )
14 cardon 7531 . . . 4  |-  ( card `  A )  e.  On
15 oncardval 7542 . . . 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 7545 . . 3  |-  ( card `  (/) )  =  (/)
19 ndmfv 5472 . . . 4  |-  ( -.  A  e.  dom  card  -> 
( card `  A )  =  (/) )
2019fveq2d 5448 . . 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 2520   (/)c0 3416   |^|cint 3822   class class class wbr 3983   Oncon0 4350   dom cdm 4647   ` cfv 4659    ~~ cen 6814   cardccrd 7522
This theorem is referenced by:  oncard  7547  cardlim  7559  cardiun  7569  alephnbtwn2  7653  infenaleph  7672  dfac12k  7727  pwsdompw  7784  cardcf  7832  cfeq0  7836  cfflb  7839  alephval2  8148  cfpwsdom  8160  gch2  8255  tskcard  8357  hashcard  11301
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 4101  ax-nul 4109  ax-pow 4146  ax-pr 4172  ax-un 4470
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 2521  df-rex 2522  df-rab 2525  df-v 2759  df-sbc 2953  df-dif 3116  df-un 3118  df-in 3120  df-ss 3127  df-pss 3129  df-nul 3417  df-if 3526  df-pw 3587  df-sn 3606  df-pr 3607  df-tp 3608  df-op 3609  df-uni 3788  df-int 3823  df-br 3984  df-opab 4038  df-mpt 4039  df-tr 4074  df-eprel 4263  df-id 4267  df-po 4272  df-so 4273  df-fr 4310  df-we 4312  df-ord 4353  df-on 4354  df-xp 4661  df-rel 4662  df-cnv 4663  df-co 4664  df-dm 4665  df-rn 4666  df-res 4667  df-ima 4668  df-fun 4669  df-fn 4670  df-f 4671  df-f1 4672  df-fo 4673  df-f1o 4674  df-fv 4675  df-er 6614  df-en 6818  df-card 7526
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