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Theorem cardidm 7592
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
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 cardid2 7586 . . . . . . . 8  |-  ( A  e.  dom  card  ->  (
card `  A )  ~~  A )
2 ensym 6910 . . . . . . . 8  |-  ( (
card `  A )  ~~  A  ->  A  ~~  ( card `  A )
)
31, 2syl 15 . . . . . . 7  |-  ( A  e.  dom  card  ->  A 
~~  ( card `  A
) )
4 entr 6913 . . . . . . . 8  |-  ( ( y  ~~  A  /\  A  ~~  ( card `  A
) )  ->  y  ~~  ( card `  A
) )
54expcom 424 . . . . . . 7  |-  ( A 
~~  ( card `  A
)  ->  ( y  ~~  A  ->  y  ~~  ( card `  A )
) )
63, 5syl 15 . . . . . 6  |-  ( A  e.  dom  card  ->  ( y  ~~  A  -> 
y  ~~  ( card `  A ) ) )
7 entr 6913 . . . . . . . 8  |-  ( ( y  ~~  ( card `  A )  /\  ( card `  A )  ~~  A )  ->  y  ~~  A )
87expcom 424 . . . . . . 7  |-  ( (
card `  A )  ~~  A  ->  ( y 
~~  ( card `  A
)  ->  y  ~~  A ) )
91, 8syl 15 . . . . . 6  |-  ( A  e.  dom  card  ->  ( y  ~~  ( card `  A )  ->  y  ~~  A ) )
106, 9impbid 183 . . . . 5  |-  ( A  e.  dom  card  ->  ( y  ~~  A  <->  y  ~~  ( card `  A )
) )
1110rabbidv 2780 . . . 4  |-  ( A  e.  dom  card  ->  { y  e.  On  | 
y  ~~  A }  =  { y  e.  On  |  y  ~~  ( card `  A ) } )
1211inteqd 3867 . . 3  |-  ( A  e.  dom  card  ->  |^|
{ y  e.  On  |  y  ~~  A }  =  |^| { y  e.  On  |  y  ~~  ( card `  A ) } )
13 cardval3 7585 . . 3  |-  ( A  e.  dom  card  ->  (
card `  A )  =  |^| { y  e.  On  |  y  ~~  A } )
14 cardon 7577 . . . 4  |-  ( card `  A )  e.  On
15 oncardval 7588 . . . 4  |-  ( (
card `  A )  e.  On  ->  ( card `  ( card `  A
) )  =  |^| { y  e.  On  | 
y  ~~  ( card `  A ) } )
1614, 15mp1i 11 . . 3  |-  ( A  e.  dom  card  ->  (
card `  ( card `  A ) )  = 
|^| { y  e.  On  |  y  ~~  ( card `  A ) } )
1712, 13, 163eqtr4rd 2326 . 2  |-  ( A  e.  dom  card  ->  (
card `  ( card `  A ) )  =  ( card `  A
) )
18 card0 7591 . . 3  |-  ( card `  (/) )  =  (/)
19 ndmfv 5552 . . . 4  |-  ( -.  A  e.  dom  card  -> 
( card `  A )  =  (/) )
2019fveq2d 5529 . . 3  |-  ( -.  A  e.  dom  card  -> 
( card `  ( card `  A ) )  =  ( card `  (/) ) )
2118, 20, 193eqtr4a 2341 . 2  |-  ( -.  A  e.  dom  card  -> 
( card `  ( card `  A ) )  =  ( card `  A
) )
2217, 21pm2.61i 156 1  |-  ( card `  ( card `  A
) )  =  (
card `  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1623    e. wcel 1684   {crab 2547   (/)c0 3455   |^|cint 3862   class class class wbr 4023   Oncon0 4392   dom cdm 4689   ` cfv 5255    ~~ cen 6860   cardccrd 7568
This theorem is referenced by:  oncard  7593  cardlim  7605  cardiun  7615  alephnbtwn2  7699  infenaleph  7718  dfac12k  7773  pwsdompw  7830  cardcf  7878  cfeq0  7882  cfflb  7885  alephval2  8194  cfpwsdom  8206  gch2  8301  tskcard  8403  hashcard  11349
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 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512
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 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pss 3168  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-tp 3648  df-op 3649  df-uni 3828  df-int 3863  df-br 4024  df-opab 4078  df-mpt 4079  df-tr 4114  df-eprel 4305  df-id 4309  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-er 6660  df-en 6864  df-card 7572
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